text:
## find lCM of 1 upon alpha + beta + 1 upon beta + gama + 1 upon gama + alpha

Question

find lCM of 1 upon alpha + beta + 1 upon beta + gama + 1 upon gama + alpha

in progress 0
1 month 2021-08-19T00:01:34+00:00 2 Answers 0 views 0

format: plain


text:
# Tikz: Using \pgfmathwidth (or \widthof) with \Large text causes crash

I;m trying to utilise this method for putting a box around text: http://tex.stackexchange.com/a/20326/17049

But it doesn't work with large text...

MWE:

\documentclass{article}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}
\pgfmathwidth{\Large "Text"}
\end{tikzpicture}
\end{document}

Causes:

! TeX capacity exceeded, sorry [input stack size=5000].
\@nomath ...e \@font@warning {Command \noexpand #1
invalid in math mode}\fi
l.6     \pgfmathwidth{\Large "Text"}

-
From the PGF Manual: "It is important to remember that any expression is expanded with \edef before being parsed, so any macros (e.g., font commands like \tt or \Huge) will need to be \protected" (e.g., \noexpand\Huge is usually sufficient)." – Herr K. Jan 23 '14 at 17:55
@KevinC With \pgfmathwidth{"\noexpand\Large Text"}, \pgfmathresult is set to 0pt. On the other hand, \pgfmathwidth{\noexpand\Large "Text"} gives errors. – egreg Jan 23 '14 at 18:01
@egreg: That's strange... I get the correct measures with your first syntax. See my answer below. – Herr K. Jan 23 '14 at 18:04

From the PGF Manual:

It is important to remember that any expression is expanded with \edef before being parsed, so any macros (e.g., font commands like \tt or \Huge) will need to be "protected" (e.g., \noexpand\Huge is usually sufficient)."

# MWE

\documentclass{article}
\usepackage{tikz}
\setlength\parskip{20pt}
\begin{document}
Width of text: \pgfmathwidth{"Text"}\pgfmathresult

Width of {\Large Text}: \pgfmathwidth{"\noexpand\Large Text"}\pgfmathresult

Width of {\Huge Text}: \pgfmathwidth{"\noexpand\Huge Text"}\pgfmathresult
\end{document}

# Output

-
Did you try it inside a tikzpicture environment? – egreg Jan 23 '14 at 18:03
@egreg: You're right, \pgfmathwidth{"\noexpand\Large Text"} doesn't work inside a tikzpicture environment, except when issued in a \node. Could this be a bug? I'm still using PGF 2.10 though. – Herr K. Jan 23 '14 at 18:14
you need two more \noexpand to make it work. there must be a duplicate for this somewhere – percusse Jan 23 '14 at 18:25
@percusse: \pgfmathwidth{"\noexpand\noexpand\noexpand\Large Text"} gives a result of 20.00003, which is the same as \pgfmathwidth{"Text"}. But this is still incorrect :( – Herr K. Jan 23 '14 at 18:34


text:
Regularization

# Label Smoothing

Label Smoothing is a regularization technique that introduces noise for the labels. This accounts for the fact that datasets may have mistakes in them, so maximizing the likelihood of $\log{p}\left(y\mid{x}\right)$ directly can be harmful. Assume for a small constant $\epsilon$, the training set label $y$ is correct with probability $1-\epsilon$ and incorrect otherwise. Label Smoothing regularizes a model based on a softmax with $k$ output values by replacing the hard $0$ and $1$ classification targets with targets of $\frac{\epsilon}{k-1}$ and $1-\epsilon$ respectively.

Source: Deep Learning, Goodfellow et al

Image Source: When Does Label Smoothing Help?

#### Papers

Paper Code Results Date Stars


text:
• BABU P D

Articles written in Pramana – Journal of Physics

• Thermal expansion behaviour and phase stability of AFe$_2$As$_2$ (A=Ca, Sr and Eu) using powder diffraction technique

The thermal expansibilities and phase stabilities of AFe$_2$As$_2$ (A = Ca, Sr and Eu) have been investigated by powder diffraction techniques in the temperature range 5–600 K.We found the anisotropic thermal expansivities with temperature for all the compounds. The lattice parameter in the tetragonal phase (AT) of CaFe$_2$As$_2$ contracts with increasing temperature, whereas CT expands. The rate of contraction in AT is lower than the rate of expansion in CT. Other compounds show normal thermal expansion behaviour along both a- and c-axes. In-plane expansion (i.e., along the a-axis) is found to be the smallest for EuFe$_2$As$_2$ and the highest for BaFe$_2$As$_2$. However, therate of change of thermal expansivities along out-of-plane (i.e., along the c-axis) is higher as we go from Ba, Sr, Eu and Ca, respectively. Above 600 K, we notice the appearance/disappearance of certain reflections which suggest that tetragonal phase is not stable above this temperature for these compounds.

• # Pramana – Journal of Physics

Volume 94, 2020
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019


text:
# prove $\sum_{k = 0}^{n} \binom{n}{k} \binom{m-n}{n-k} = \binom{m}{n}$

prove $\sum_{k = 0}^{n} \binom{n}{k} \binom{m-n}{n-k} = \binom{m}{n}$

Attempt:I was thinking of trying to prove this through induction, but I am having trouble with a base case:

base case: let $n = 2$: $$LHS = \binom{2}{0} \binom{m-2}{2} + \binom{2}{1} \binom{m-2}{1} + \binom{2}{2} \binom{m-2}{0} \\ = \frac{(m-2)[(m-3) + 2!2!(m-3)!] + 2!}{2!} \ (after\ simplification)$$

$$RHS = \frac{m!}{2!(m-2)!}$$

But I am stuck as what to try next to at least equate these two expressions.

Note: I took a look at Vandermonde's identity on wikipedia and the ensuing proof, but the proof still leaves out how to make the transition between the two initial expressions

• What seems to be the problem in using Vandermonde's idendity? Should work is it not? – Satish Ramanathan May 27 '16 at 17:09

Use Pascal’s identity a few times:

\begin{align*} \binom20\binom{m-2}2+\binom21\binom{m-2}1+\binom22\binom{m-2}0&=\binom{m-2}2+2\binom{m-2}1+\binom{m-2}0\\ &=\left(\binom{m-2}2+\binom{m-2}1\right)+\\ &\qquad\qquad\left(\binom{m-2}1+\binom{m-2}0\right)\\ &=\binom{m-1}2+\binom{m-1}1\\ &=\binom{m}2\;. \end{align*}

But why start at $n=2$? Starting at $n=0$ makes the base case very simple.

• I was curious to see the expressions in "action", like you said $n = 0$ is easy, but you don't get to see anything really going on. – dc3rd May 27 '16 at 20:55

A much easier approach would be to look at both sides as the coefficient of $x^n$ in the binomial expansion of $(1+x)^m$.

The right hand side follows immediately.

For the left hand side, write $(1+x)^m$ as $(1+x)^n$$(1+x)^{m-n}$ and use the binomial theorem for each term in the product individually.

The coefficient for $x^n$ in this case is precisely the expression on the left hand side.

Combinatorics proof: From a group of $m$ people (among which $n$ are left-handed) we can pick a committee of $n$ people by picking $k$ left-handers and $n-k$ right-handers, $0\le k\le n$.

\begin{align} \color{#f00}{\sum_{k = 0}^{n}{n \choose k}{m - n \choose n - k}} & = \sum_{k = 0}^{n}{n \choose k}\ \overbrace{\oint_{\verts{z} = 1}{\pars{1 + z}^{m - n} \over z^{n - k + 1}} \,{\dd z \over 2\pi\ic}}^{\ds{{m - n \choose n - k}}}\ =\ \oint_{\verts{z} = 1}{\pars{1 + z}^{m - n} \over z^{n + 1}}\ \overbrace{\sum_{k = 0}^{n}{n \choose k}z^{k}}^{\ds{\pars{1 + z}^{n}}}\ \,{\dd z \over 2\pi\ic} \\[3mm] & =\ \underbrace{% \oint_{\verts{z} = 1}{\pars{1 + z}^{m} \over z^{n + 1}} \,{\dd z \over 2\pi\ic}}_{\ds{{m \choose n}}}\ =\ \color{#f00}{{m \choose n}} \end{align}

• I believe you were one of the first to present the Egorychev method at MSE. (+1). – Marko Riedel May 28 '16 at 22:13
• @MarkoRiedel I don't know it's true. I was not aware it has a name. Fine to know that. Thanks. – Felix Marin May 28 '16 at 22:17
• If you have access to a university library you may consult "Integral representation and the computation of combinatorial sums" by G. P. Egorychev, translation published by the AMS. – Marko Riedel May 28 '16 at 22:22
• @MarkoRiedel Thanks. Maybe, I can buy it at Amazon. I learnt many tricks with Donald Knuth ( TeX author ) books. – Felix Marin May 28 '16 at 22:34
• Clearly we may suppose that D. Knuth deploys this method as it is basically the Cauchy Residue Theorem, but I believe it was Egorychev who wrote a book about it and put its detailed workings on an algorithmic footing. Many well-known mathematicians have used complex variables in combinatorics. – Marko Riedel May 28 '16 at 22:39


text:
# Question #05bea

Mar 11, 2017

$\frac{1}{\frac{2}{6}} = 1 \times \frac{6}{2} = 3$

$2$ sixths make a third

$\frac{\frac{1}{3}}{\frac{1}{6}} = 2$

#### Explanation:

For the first one and the third one, use the same concept of being able to flip the denominator if it was a fraction and change the question to multiplication.

For the second one, what you do is multiply $\frac{1}{6}$ by $2$ to get $\frac{1}{3}$

$\frac{1}{6} \times 2 = \frac{2}{6} = \frac{1}{3}$

Mar 11, 2017

$1 \div \frac{2}{6} = 3$

It takes two sixths to make $\frac{1}{3}$

$\frac{1}{3} \div \frac{1}{6} = 2$

#### Explanation:

Using the shortcut method

Part 1: $\to 1 \div \frac{2}{6}$

So that you fully understand what is going on write 1 as $\frac{1}{1}$

$\frac{1}{1} \div \frac{2}{6}$ turn the $\frac{2}{6}$ upside down and multiply

$\frac{1}{1} \times \frac{6}{2} \text{ "=" } \frac{1 \times 6}{1 \times 2} = \frac{6}{2}$

$\frac{6}{2} = \frac{6 \div 2}{2 \div 2} = \frac{3}{1} = 3$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Part 2:$\to \text{ how many sixths make } \frac{1}{3}$

Let the unknown number be $x$

$\frac{x}{6} = \frac{1}{3}$

Multiply by 1 and you do not change the value. However, 1 comes in many forms so you can change the way a value looks without changing 'what it is worth'.

$\textcolor{g r e e n}{\frac{x}{6} = \frac{1}{3} \textcolor{red}{\times 1}}$

$\textcolor{g r e e n}{\frac{x}{6} = \frac{1}{3} \textcolor{red}{\times \frac{2}{2}}}$

$\frac{x}{6} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}$

So it takes two sixths to make $\frac{1}{3}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Part 3:$\to \text{What is } \frac{1}{3} \div \frac{1}{6}$

Using the shortcut method of turning upside down and multiplying

$\frac{1}{3} \div \frac{1}{6}$ gives the same answer as $\frac{1}{3} \times \frac{6}{1}$

$\frac{1 \times 6}{3 \times 1} = \frac{6 \div 3}{3 \div 3} = \frac{2}{1} = 2$


text:
The 2014 Nordic Collegiate Programming Contest

Start

2014-10-04 11:00 CEST

The 2014 Nordic Collegiate Programming Contest

End

2014-10-04 16:00 CEST
The end is near!
Contest is over.
Not yet started.
Contest is starting in -1475 days 22:15:56

5:00:00

0:00:00

Problem KTrain Passengers

The Nordic Company of Passing Carriages is losing money at an alarming rate because most of their trains are empty. However, on some lines the passengers are complaining that they cannot fit in the cars and have to wait for the next train!

The authorities want to fix this situation. They asked their station masters to write down, for a given train, how many people left the train at their station, how many went in, and how many had to wait. Then they hired your company of highly paid consultants to assign properly sized trains to their routes.

You just received the measurements for a train, but before feeding them to your optimisation algorithm you remembered that they were collected on a snowy day, so any sensible station master would have preferred to stay inside their cabin and make up the numbers instead of going outside and counting.

Verify your hunch by checking whether the input is inconsistent, i.e., at every time the number of people in the train did not exceed the capacity nor was below $0$ and no passenger waited in vain (i.e., waited on the station when there was room in the train). The train should start and finish the journey empty, in particular passengers should not wait for the train at the last station.

Input

The first line contains two integers $C$ and $n$ ($1 \leq C \leq 10^9$, $2 \leq n \leq 100$), the total capacity and the number of stations the train stops in. The next $n$ lines contain three integers each, the number of people that left the train, entered the train, and had to stay at a station. Lines are given in the same order as the train visits each station. All integers are between $0$ and $10^9$ inclusive.

Output

One line containing one word: possible if the measurements are consistent, impossible otherwise.

Sample Input 1 Sample Output 1
1 2
0 1 1
1 0 0

possible

Sample Input 2 Sample Output 2
1 2
1 0 0
0 1 0

impossible

Sample Input 3 Sample Output 3
1 2
0 1 0
1 0 1

impossible

Sample Input 4 Sample Output 4
1 2
0 1 1
0 0 0

impossible


text:
DesignWIKI

Fil Salustri's Design Site

COLLECTIONS

2015.01.09 11:13

How to use EXCEL to quickly calculate letter grades from percentage grades.

While most instructors assign grades on a scale out of 100, letter grades are reported to the Registrar. Assuming you use D2L or Excel to track grades, there's an easy way to have Excel generate the letter grades that correspond to given percentage (or any other) grades.

Using D2L

D2L can't do the conversion. But you can download an excel-compatible form of the grade center from D2L.

• Enter your D2L course shell.
• Go to the Grade Center.
• Above the grades table, to the right, you will see a button marked Work Offine.
• Select the Download option from Work Offline.
• You will be presented with a screen of options. The options are fine as they are.
• Click Submit.
• On the next screen, click Download.

Using Excel

We assume:

• you have an excel spreadsheet with one row per student, and
• each row has one cell containing the percentage grade for that student.

Somewhere in the spreadsheet, you must create a table that has two columns.

• The first column contains the letter grades.
• The second column contains the lowest percentage grade for the corresponding letter grade.

Here's an example:

F 0
D- 50
D 53
D+ 57
C- 60
C 63
C+ 67
B- 70
B 73
B+ 77
A- 80
A 85
A+ 90

Next, you use the excel LOOKUP function to map a given grade to a letter grade.

The general form of the LOOKUP function is:

LOOKUP ( Student grade cell, Range of cutoffs, Range of letter grades )

Remember to use dollar signs to fix the location of the range of cutoffs and range of letter grades.

• For example, if you've used columns A and B, starting at row 1, for the table above, you could write the following:
LOOKUP ( C16, $B$1:$B$13, $A$1:$A$13 )

You can write this formula for only one cell, then copy and paste it into the rest of the cells.

If you're using Google Spreadsheets, you need to do it slightly differently. Google doesn't support the LOOKUP function - indeed, it remains in Excel for backward compatibility only.

The difference is minor: you just need to use a different equation. Instead of LOOKUP, use this:

INDEX ( range of letter grades, MATCH ( student grade cell, range of cutoffs ))

You may have a student grade of 89.7 which you want to turn into A+. The above technique will not do this because 89.7 is still less than the cutoff for A+.

To fix this, you need to ROUND the percent grade before doing the lookup. Here's the same example with the rounding operation included.

LOOKUP ( ROUND (C16,0), $B$1:$B$13, $A$1:$A$13 )

<refnotes> notes-separator: none </refnotes>


text:
# Learning by Gradient Descent: Stochastic Approximation

Learning with Machines on large multidimensional datasets can be computationally intensive especially if significant batches of data samples are necessary or used for each learning step - as in batch gradient descent for perceptron training. However, if at each learning step, the learning is not differentiable, we end up with an iteration of next to similar steps on the gradient slope which do not significantly say a lot about the learning machine.

Let’s first understand what Gradient descent is, variations of it and how stochastic approximation can come into play. Using a simple explanation given in the Deep Learning Book(Ian Goodfellow et al);

Consider an objective function $$f(x)$$ that we want to optimize i.e. minimize or maximise it by altering x.

We obtain its derivative $$f^1(x)$$ which gives the slope of $$f(x)$$ at point x, thus we can scale a change in input to obtain a corresponding change in output. For examples $$f(x +\in) \sim f(x) + \in f^1(x)$$. Therefore, to minimize the function $$f(x)$$, we can move small steps in the opposite direction of the derivative $$f^1(x)$$. That is basically gradient descent.

The objective function and its derivative values represent crucial parts of the gradient learning ; $$f^1(x) = 0$$, which are referred to as critical or stationary points which provide no information about direction of gradient and these points fall within the following categories;

• Local minima- refers to a point lower than all neighboring points
• Local Maxima - refers to a point higher than all neighboring points
• Saddle points - Neither a maxima or minima
• Global Minima - is the lowest value of f(x) throughout the input space and it is highly possible for a local minima not to be globally optimum

Beyond and/or above the local maxima and minima it is not possible to increase and/or decrease the value of $$f(x)$$.

The general Gradient Update rule can be defined as follows;

$W_{t+1} = w_t - \eta \bigtriangledown E\mid w_t$

In this case, we consider perceptron learning of weights(Haykin has a great explanation) over $$t$$ learning steps and $$E$$ is the cost function. Therefore the next weight $$w_{t+1}$$ can be estimated by the current $$w_t$$ minus a product of the learning rate $$\eta$$ and a derivative of the cost function $$E$$ w.r.t the weight.

The negative gradient gives the direction of the steepest descent in E.

Further we will explore the types of gradient descent; Batch which was mentioned earlier and online learning which processes each observation, one at a time, updating the gradient after learning step. Online learning has several advantages over the Batch learning i.e no need to store complete dataset, easy adaptability in cases of changing data and many more.

Learning gradient descent with stochastic approximation is basically online learning but the singular examples are selected randomly with equal probability at each learning update for $$tmax \cdot P$$ learning steps(P is the number of observations in the dataset). Therefore the true gradient of the cost function $$E$$ is approximated by a single example.

Below are my Matlab code samples that can be refactored into any programming langauge.

sgd.m

cost.m

The cost function and generalisation error values can be calculated after $$P$$ randomised steps to monitor the learning performance. As seen below, we can see plateau states in the cost function after certain learning steps. At these points there isn’t much learning happening therefore the generalisation error will significantly increase. This behavior can also explain poor generalisation and overfitting at the plateau states.

Figure shows plateau states from Stochastic Gradient descent

Figure shows final weights from Stochastic Gradient descent. The weights signify the importance of the features

-Deep learning by Ian Goodfellow, Yoshua Bengio, Aaron C. Courville

-The Elements of statistical learning by Jerome H. Friedman, Robert Tibshirani, and Trevor Hastie

-Neural networks and learning machines by Simon S. Haykin


text:
I tried to solve different tasks, but I was not so successful. This write-up is complete from the point of view of the solution… but I didn’t managed to get the flag.

So why am I posting it?

Because I think this is worth sharing. Be patient and keep reading ;-)

The challenge is accessible via a zip folder. The password of that zip is the flag of a previous challenge. The main contents of the folder are the following files:

• a lore for the challenge;
• examples.p: a pickle file containing, for each row a tuple of five floats and a couple of int;
• page_of_numbers.p: a pickle file containing, for each row a tuple of five floats.

Mmmh. What the hell? Let’s read the lore. Be careful in reading this… because this lore matters. You’ll understand why at the bottom of this write-up.

The acolytes of several monasteries have been studying possible origins of this attack for weeks.
One of the most ancient and forbidden tomes contained in the great libraries talks about the origin of the purple dragon guard.
It appears that in ancient times, long before spellscourge, they were formidable gate keepers of Cormyr to other planes.
The most renowned branch of them fought back the great black dragon Thauglorimorgorus who was so ancient his scales turned purple.
They ended up chasing it into the Vast Swamp and founded an encampment called The Swamp Fort of Cormyr.
In this fort was a portal to the shadow plane to where they had chased Thauglor.
Those who entered never returned.  It appeared Thauglor had never been pushed back but all along was luring the militia to slaughter.
In another expedition through the portal, desperate times forced a cleric to ask a miracle of Selûne.
Our lady of silver had always been at odds with her twin, Shar, the mistress of night goddess of the plane of shadow.
The call was answered and a counterpart of this fort sprang into being with a pillar of light which extinguished the magical darkness
for miles all around.
Thauglor was repelled by this brilliant light.
This lead to many victories in this plane.  The guard resided here with the cleric at its head named The Lord of Light.
She thus forward sat on the crystal throne and lead the guardians of this shining citadel known to all as The Ever Shining Army.
This appears to be solely a legend since the only known encampment in the Vast Swamp is known as the Lost Refuge
and is rumored to house those of Church of Shar, disciples of the shadow weave.
They are nothing like those known in that legend.
It is however known that there is a fortress on the other side ruled by the Dusk Lord known as the Shadow Citadel which is situated
on the edge of the Shadow Swamp.
The mountains there are treacherous and the toxic rivers which run through the valleys sometimes lead historians to call it The Vale of Shadows.
One cleric has spent time divining a spell which determined that those pages of numbers previously found are trajectories
of the format <x position, y position, speed, angle above the horizontal, direction from the firing position>.
It appears to be that there is a spell being cast using the shadow weave and the coordinates seem to indicate
that the spells location is in the kingdom of Cormyr.

Basically, each tuple corresponds to a bullet! I have a (x,y) position, the speed of shooting, the angle above the horizontal and direction from shooting position. It’s simple 3D physics: I need to calculate the trajectory of the bullet and it’s impact point.

SPOILER: I was given a little aid by my dear brother Andrea (who is a Physicist). I learnt something new while reasoning with him. I’m not talking about calculus, formulae or physics laws, but lateral thinking in general. Having said that, let’s do some algebra.

# The physics behind this challenge

I’m going to refer to $(x,y)$ for the position, $v$ for speed, $\theta$ for angle above the horizontal and $\alpha$ for the direction. First of all, I need to understand how much time the bullet needs to fall to the ground again. I can state that:

$$v_z = v_{z0} - |g|t$$

When it will reach the ground? On the vertex of the parabule, $v_z$ is zero, then $v_z$ must be the opposite of $v_{z0}$ when it will touch the ground. It’s not an out-of-the-blue result, I just used the law of conservation of energy: the sum of kinetic and potential energy must be constant (there is no air friction). There will be only potential energy in the vertex of the parabule, while it is only kinetic energy at the beginning and the end of the motion.

$$v_z = -v_{z0} \\ -v_{z0} = v_{z0} - |g|t\\ t = \frac{2v_{z0}}{|g|}$$

I have now when the bullet impacts the xy plane. I need now to calculate $v_{z0}$:

$$v_{z0} = vsin(\theta)$$

It’s time to calculate how long the bullet will travel on x and y.

I need to decompose the speed, projecting it on the $xz$ and $yz$ plane:

$$v_x = vcos(\theta)cos(\alpha)\\ v_y = vcos(\theta)sin(\alpha)\\$$

Now that I have the projections of the speed, I can calculate the impact position. I am going to use again the impact time calculated before:

$$x_f = x + v_x * t \\ y_f = y + v_y * t$$

# It’s time to Fail

Even tough the model is correct, I couldn’t match the results of examples.p file. The model calculates the trajectory of a bullet from a point $(x,y)$ in space to $(x_f, y_f)$.

## IN FACT IT WAS THE CONTRARY.

It is written clearly in the lore:

<... direction from the firing position>

So, the $(x,y)$ couple is an impact point, not a shooting point. Signs need to be flipped here:

$$x_f = x - v_x * t \\ y_f = y - v_y * t$$

… leading to this flag. You can find a simple script which calculates trajectory for all the points.

def predict_impact_point(row):
x,y = row[0],row[1]
v = row[2]
theta = row[3]
alpha = row[4]
vx = v * np.cos(theta)*np.cos(alpha)
vy = v * np.cos(theta)*np.sin(alpha)
vz = v * np.sin(theta)
g = 9.81
t = 2*vz / g
return x - vx*t, y - vy*t

matrix_data = [ ]
res = []
for d in data:
val =predict_impact_point([i for i in d])
matrix_data.append(val)
matrix_data = np.array(matrix_data)

plt.plot(matrix_data[:,0], matrix_data[:,1],'o')
plt.show()

(SPOILER: there is a last trick, but I’ll let you discover by youself…)

We didn’t managed to capture the flag but we were so damn close. We had fun, that was the most important thing. Here you can download the points and script I used to solve the challenge.

See you next write-up ;)


text:
# Pay-to-play Mining training

This guide describes the most effective pay-to-play methods to train the Mining skill. Mining is a gathering skill which typically involves collecting ores from different types of ore rock; ore rocks are often found in groups at mining sites. Players can also occasionally obtain uncut gems and geodes while mining. A pickaxe is required when mining. Levelling up the Mining skill increases the chance of the player getting critical hits, which deal greater amounts of damage to rocks and thereby increase the rate of receiving ores and larger experience drops. A higher Mining level gives chances of receiving double ores from certain types of rocks and increases the frequency of rockertunities, which also grant extra damage when used. As players level up, they are also able to use better pickaxes. To gain the best experience rates possible, players are strongly recommended to use the highest level pickaxe available to them and rockertunities whenever available.

A rockertunity

The experience rate estimates in this guide assume that the best possible pickaxe is being used for the stated Mining level and do not incorporate the use of any experience boosting items or bonus experience. They also assume that players do not bank the ores that they obtain. Banking ores decreases the amount of experience players can gain per hour and, as most ores are not very valuable, is usually not worth the time taken. Players using the Evolution of Combat mode can drop ores by adding the ore to their action bar and pressing the associated keybind to drop the ore. As opposed to dropping items directly from the inventory using their right-click drop option, dropping ores via the action bar does not interrupt mining. Players who do wish to keep the ores that they mine could make use of the ore box and expansions to its capacity to decrease the frequency of trips to the bank. Signs of the porter could also be used to send ores to the bank as they are mined.

## Pickaxes

Higher level pickaxes deal greater amounts of damage to rocks and thereby increase the rate of receiving ores and larger experience drops. Pickaxes may be used from the player's inventory or tool belt or whilst being equipped. This guide assumes that the best possible pickaxe is being used.

Level required Pickaxe Average damage done to rock
1 Bronze pickaxe 5
10 Iron pickaxe 10
Iron pickaxe + 1 11
20 Steel pickaxe 20
Steel pickaxe + 1 21
30 Mithril pickaxe 30
Mithril pickaxe + 1 31
Mithril pickaxe + 2 32
50 Rune pickaxe 50
Rune pickaxe + 1 51
Rune pickaxe + 2 52
Rune pickaxe + 3 53
60 Orikalkum pickaxe 60
Orikalkum pickaxe + 1 61
Orikalkum pickaxe + 2 62
Orikalkum pickaxe + 3 63
Dragon pickaxe 63
70 Necronium pickaxe 70
Necronium pickaxe + 1 71
Necronium pickaxe + 2 72
Necronium pickaxe + 3 73
Necronium pickaxe + 4 74
Crystal pickaxe 74
80 Bane pickaxe 80
Bane pickaxe + 1 81
Bane pickaxe + 2 82
Bane pickaxe + 3 83
Bane pickaxe + 4 84
Imcando pickaxe 84
90 Elder rune pickaxe 90
Elder rune pickaxe + 1 91
Elder rune pickaxe + 2 92
Elder rune pickaxe + 3 93
Elder rune pickaxe + 4 94
Elder rune pickaxe + 5 95
Pickaxe of Earth and Song 95

### Augmented pickaxes

Players with at least level 22 Invention can discover how to augment pickaxes. This allows the player to create augmented crystal pickaxes, augmented Imcando pickaxes, and augmented Pickaxes of Earth and Song by using augmentors on regular pickaxes. At the same Invention level, players can discover how to make tool gizmos. A tool gizmo may be added to an augmented pickaxe to give it perks. The following perks may be useful to players training Mining:

Recommended perks
Perk Icons Perks Effect
Honed 6 Has a 12% chance of giving double ores.
Furnace 4 + Fortune 3 Has a 20% chance of granting double experience, without any progress.
Has a 1.5% chance to double and bank what you have gathered.
Possible perks
Name Effect Number of ranks
Furnace Has a 5% chance per rank of granting double experience, without any progress. 4
Honed Has a 2% chance per rank of giving double ores. 6
Imp Souled Has a 3% chance per rank to bank what you have gathered for the cost of 30 prayer points. 6
Polishing Has a 3% chance per rank of transmuting a gathered resource to a higher tier. 4
Refined Increases the chance of finding a geode by 2% per rank. 4
Wise When equipped, +1% per rank additional experience, up to 50,000 experience per day. 4
Ancient invention
Fortune Has a 0.5% chance per rank on gathering skills and 0.1% chance per rank on production skills to double and bank what you have gathered or produced. 3

To use an augmented pickaxe's perks, the player must have charges in their charge pack. Use of charges can be reduced by researching charge drain improvements or by adding the efficient perk to the item.

The Wise perk can also be put on an off-hand weapon or shield, and it will give the bonus experience regardless, enabling players to have two useful perks on their pickaxe in combination with their skilling outfit.

## Other useful equipment

### Ore box

Ore boxes are items that store metal ores. They can store up to 140 of each ore when fully upgraded. Only one ore box can be used at a time. The ore box can be added to the action bar, and clicking it there allows filling of the box without interrupting mining.

### Varrock armour

Different tiers of Varrock armour are awarded on completion of the easy, medium, hard, and elite sets of Varrock achievements. Varrock armour is a body slot item. While mining with the armour equipped, players have a chance of mining two ores at the same time and receiving Mining experience for both of them. Higher tiers of the armour grant this effect at rocks which have a higher Mining level requirement to mine, in addition to having the effects of the lower tier verions.

Players who have started the Dimension of Disaster quest can complete the easy New Varrock Achievements set to gain access to the armour's benefits without having it equipped. Completion of the medium set doubles the effects of the Varrock armour.

Item Ore rocks affected Effect Effect with New Varrock medium tasks
Varrock armour 1
• Copper
• Tin
• Iron
• Coal
4% 8%
Varrock armour 2
• Mithril
• Luminite
3% 6%
Varrock armour 3
• Runite
• Orichalcite
• Drakolith
• Necrite
• Phasmatite
2% 4%
Varrock armour 4
• Banite
• Light animica
• Dark animica
1% 2%

### Golden mining suit

Wearing pieces of the golden mining suit increases base experience gained while training Mining. Items from the set can be found when mining at the Lava Flow Mine. Players need to deal 65,000 mining damage to the crust to disturb a liquid gold nymph who can be spoken to in order to gain a part of the golden mining suit.

If a player has Varrock armour as well as all of the pieces of the golden mining suit, then players will receive the effects of the Varrock armour while wearing a complete golden mining suit.

Image Name Experience boost
Golden mining helmet 1%
Golden mining top 1%
Golden mining gloves 1%
Golden mining trousers 1%
Golden mining boots 1%
Set bonus 1%
Total experience boost 6%

### Golem outfits

Parts of golem outfits could be won as a prize from Treasure Hunter between the 19 June and 23 June 2014. The sapphire golem outfit, emerald golem outfit and ruby golem outfit consist of head slot, body slot, leg slot, hand slot and feet slot items. When wearing a complete version of any of these three outfits, players gain the following bonuses:

• 5% increased critical hit chance.
• If a player owns a complete golden mining suit, they will receive its 6% increased base experience effect while wearing a golem outfit.
• If a player owns Varrock armour, they will receive its chance of receiving two ores at once effect while wearing golem outfit.

Players who have all pieces of the sapphire golem outfit, emerald golem outfit, and ruby golem outfit can combine these outfits' components to form the magic golem outfit. This outfit gives superior benefits to those given by the other golem outfits. In addition to those mentioned above, players gain the following bonuses from wearing a magic golem outfit:

### Mining cape

The mining cape may be purchased by players who have achieved level 99 Mining. When equipped, the cape gives a 5% chance of receiving double ores on core rocks, and being awarded bonus experience equal to the level requirement for mining that type of rock. Players who own a max cape, completionist cape, or trimmed completionist cape may use the mining cape on one of these capes to give it the mining cape's perk.

### Starfury outfits

Parts of starfury outfits could be made by combining starfire outfit pieces with starborn diamonds, available from Treasure Hunter between the 9 September and 14 September 2015. When wearing starfury outfit pieces players have a 1% extra critical hit chance per piece of the armour equipped (excluding the cape) while mining seren stones. The benefits of the corresponding pieces of the golden mining suit also apply if they are owned.

### Brooch of the Gods

The Brooch of the Gods is worn in the pocket slot. It can store up to 1000 decorated urns and has the mining-specific benefit of increasing rockertunity multiplier by 1. A divine blessing can also spawn while mining, granting invention components.

### Summoning familiars

There are five Summoning familiars available which add a boost to the damage dealt per swing while mining when they are summoned.

Familiar GE cost of pouch Summoning level required Damage boost Familiar duration (minutes)
Desert wyrm 824 18 1 19
Void ravager 10,791 34 1 27
Obsidian golem 3,659 73 7 55
Gargoyle (familiar) 47,468 75 12 64
Lava titan 4,269 83 10 61

The special move of the Gargoyle, Hammer Rock also allows your next mining swing to deal 50% more damage.

### Waterfiend (familiar)

The waterfiend is a summoning familiar that grants a 5% chance while training gathering skills (including Mining) to duplicate the gained resource and store it within itself. The familiar can carry up to 20 generated items.

Straight Flush scroll allows you to teleport all items stored within the familiar's inventory to your bank at the cost of 6 Special move points.

### Urns

Urns can be used to gain additional Mining experience while training. When held in the inventory, urns are filled as the player mines ore. Only base experience gains counts towards filling the urn. Full urns can be teleported away to receive additional experience, equivalent to 20% of the Mining experience gained while filling the urn. Only one urn is filled at a time, though players may own up to a total of 10 full urns. If a player owns 10 full mining urns they must teleport at least one away before they can fill more urns. Urns are only tradeable in their (nr), non-runed, forms. In order to use a mining urn the player must add an earth rune to it, this makes the urn untradeable.

Image Name Crafting level required to add rune GE price of non-runed urn Charges gained from Type of ore Experience required to fill Experience gained from teleporting GP/XP (Including rune)
Cracked mining urn 2 1,532 Ores available at level 1 Clay
Copper ore
Tin ore
Rune essence
437.5 87.5 17.7
Fragile mining urn 17 1,099 Ores up to and including level 20 The above, plus:
Iron ore
Limestone
Blurite ore
Coal
Silver ore
Daeyalt ore
1,000 200 5.58
Plain mining urn 32 1,660 Ores up to and including level 50 The above, plus:
Mithril ore
Pure essence
Sandstone
Luminite ore
Gold ore
1,625 325 5.16
Strong mining urn 48 2,507 Ores up to and including level 70 The above, plus:
Granite
Runite ore
Orichalcite ore
Drakolith ore
Necrite ore
Phasmatite ore
2,000 400 6.31
Decorated mining urn 78 4,784 Ores up to and including level 98 The above, plus:
Living minerals
Bane ore
Red sandstone
Seren stone
Salty crabletine
Salty crablet
Alaea crablet
Idol crablet
3,125 625 7.68

### Urn enhancer

Players who have at least level 24 Invention and who have completed Nomad's Elegy can discover how to make an urn enhancer. When in the player’s inventory the urn enhancer grants 25% additional experience for teleporting a full urn. This effect costs 20 charges from the player’s charge pack. Assuming the player purchases divine charges from the Grand Exchange, it would cost 638.34 to use the urn enhancer when teleporting an urn. A player mining granite using light form and crystallise with level 99 Mining would gain around 169,500 base experience an hour. If using decorated mining urns the player would gain 33,900 extra experience an hour, with an additional 8,475 experience if using an urn enhancer. This would cost around 34,623.56 an hour in divine charges.

### Mining accumulators

Mining accumulators can be created by players with at least level 50 Invention, for 25 dwarven currency if the player has chosen a technology tree. They can also be bought from the Grand Exchange for 38,341 each. When held in the player’s inventory a Mining accumulator gains 1% charge each time that the player has a critical hit while mining. The accumulator can then be studied to gain experience. 2,500 Mining experience is gained from studying a fully charged Mining accumulator. If Mining accumulators are bought on the Grand Exchange, this gives them a cost of 15.34 coins/XP.

### Rock-crushing scrimshaw

Rock-crushing scrimshaws can be activated to give the player an increased critical hit chance. Rock-crushing scrimshaws can be activated and deactivated at will, providing that a 10 second cooldown has passed. The inferior rock-crushing scrimshaw gives a 2% increased critical hit chance and requires level 70 Mining to use. The inferior scrimshaw is tradeable, costing 1,178,847, and lasts for three hours. The superior rock-crushing scrimshaw gives a 3% increased critical hit chance and requires level 80 Mining to use. The superior scrimshaw lasts for four hours but it is untradeable. Players who have level 85 Fletching can unlock the ability to make superior rock-crushing scrimshaws by gathering all four fragments of the rock-crushing scroll in the Player-owned port minigame. The superior scrimshaw can then be crafted from four ancient bones.

### Stone spirits

Stone spirits double every ore mined, in exchange for one stone spirit of the ores type. No additional experience is awarded for the extra ore.

When combined with a sign of the porter, one spirit is consumed, one porter charge is consumed, and two ores are banked.

### Juju potions

Juju mining potion adds a 10% chance to gain a stone spirit of the type of ore mined. Perfect juju mining potion and Perfect plus potion have the same effect, and also completely fill the mining stamina bar when a stone spirit is consumed.

### Powerburst of opportunity

The powerburst of opportunity is a powerburst potion that, for the next 15 seconds after use, doubles the effect of rockertunities. This means that the rockertunity damage multiplier is doubled. Powerburst potions have a cooldown of two minutes.

### Torstol incense sticks

Each potency level of torstol incense sticks provides a 0.5% increase to base XP gain. The potency level will increase every 10 minutes and will reach its maximum at level 4 (+2% to base XP). Overloading consumes 6 sticks but instantly provides the maximum potency.

### Auras

Quarrymaster auras increase a player's critical hit chance while they are activated. These auras' effects last for one hour after they are activated. Players may use five or ten vis wax to extend the quarrymaster aura's activation time by 50% or 100% (costing 61,675 or 123,350). This effect must be applied before the aura is activated. Once a quarrymaster aura has been used the player must wait three hours before using it again. Quarrymaster auras can be bought using Membership loyalty points. All lower tiers of the aura must be owned before a higher tier aura can be purchased.

Tier Percentage chance of getting critical hit increased by Loyalty point cost
Individual Cumulative
1 Quarrymaster 2% 5,000 5,000
2 Greater quarrymaster 4% 14,000 19,000
3 Master quarrymaster 6% 33,500 52,500
4 Supreme quarrymaster 8% 58,500 111,000
5 Legendary quarrymaster 10% 119,000 230,000

Resourceful aura gives the following mining benefits:

• +2% geode chance when mining.
• 10% chance to give an additional red/crystal-flecked sandstone which does not contribute towards the daily limit.

### Skillchompas

Skillchompas are items that can be used from the player's inventory while mining. While mining skillchompas give additional damage on critical hits, but are consumed in the process. The Player-owned farm perk "If Chins Could Skill", activated by having one/two small pens with a totem and at least one elder chinchompa, will cause skillchompas to not be consumed 3%/5% of the time.

Image Name Mining level required Bonus critical hit damage GE price per skillchompa
Cobalt skillchompa 31 30 515
Viridian skillchompa 41 40 520
Azure skillchompa 51 50 531
Crimson skillchompa 61 60 729
Crystal skillchompa 71 70 963

### Crystallise and Light Form

A crystallised skilling node

Access to Seren spells and prayers, which can be found in the ancient magicks and ancient curses spell and prayer books, is unlocked upon completing The Light Within quest. Level 75 Mining is required for completion of the Plague's End quest, which is one of the requirements needed to complete The Light Within. Crystallise requires level 88 Magic and six water, six fire, six chaos and six soul runes to cast.

Crystallise can be cast on a rock to increase the Mining experience gained from it by 20%, while preventing the player from gaining any progress or ore. Crystallise may be cast on a rockertunity rock before it is mined and will stack with the rockertunity effect. Crystallise lasts for 30 seconds after it has been cast; it is not possible to cast Crystallise again until the effect has worn off. If a crystallised rock is being mined only by the player who cast Crystallise on it then it will not deplete until Crystallise's effect wears off. With at least level 80 Prayer players can use the Light Form prayer, which enhances the effects of Seren-aligned spells while it is activated, to further increase additional Mining experience gained from a crystallised ore rock to 40%. Players will need to consume around 53 doses of prayer potion per hour, with level 99 Prayer, to maintain the prayer points required to keep Light Form activated. Use of prayer renewal potions can help to regain some prayer points. When training Mining using Crystallise players can cast the spell up to 120 times an hour, costing 2,280,240 in runes. This cost can be reduced by using staves to provide some of the required runes.

Crystallise can only be cast on core rocks: copper, tin, iron, mithril, adamantite, luminite, runite, orichalcite, drakolith, necrite, phasmatite, banite ore rocks, light animica and dark animica.

### Luck Enhancers

By default, all base ores from level 60 have a 1% chance of awarding a metamorphic geode in place of an igneous geode whilst mining. Luck enhancers increase the odds of receiving a metamorphic geode. Wearing a charged Amulet of glory increases the chance of finding geodes by 1%.

Luck Tier Increase Total Chance
1 +0.5% 1.5%
2 +1% 2%
3 +1.5% 2.5%
4 +2% 3%

### Strength Enhancing Potions

As 1/10 of the player's Strength level is added to the player's progress gains, stat boosting potions that increase Strength can increase the efficiency that you mine.

Potion Boost
Strength potion 1 + 8%
Super strength potion 2 + 12%
Grand strength potion 2 + 14%
Extreme strength

3 + 15%
Supreme strength potion

4 + 16%

### Dragon battleaxe

The special attack can be activated in legacy every 5 minutes without needing to gain adrenaline in combat, thus becoming a free and reusable boost to Strength that increases damage done to rocks whilst mining. The total Strength level boost can then be determined:

${\displaystyle {\text{Strength level boost}}=10+{\bigg \lfloor }{\frac {5+5+5+11}{4}}{\bigg \rfloor }=10+{\bigg \lfloor }{\frac {26}{4}}{\bigg \rfloor }=10+\lfloor 6.5\rfloor =10+6=16}$

And given that in this example the player has level 99 Strength, the boosted strength level is:

${\displaystyle {\text{Boosted Strength level}}=99+16=115}$

### Relic powers

Players with level 117 Archaeology can activate the Inspire Effort relic power at the mysterious monolith for a 2% experience increase to Mining.

### Asciatops Perk

The asciatops perk from Anachronia Dinosaur Farm increases critical hit chance while mining by 3% (Tier 1) and 6% (Tier 2). Requires 110 Farming.

### Scimitops Perk

The scimitops perk from Anachronia Dinosaur Farm increases the chance of finding a geode while mining by 1% (Tier 1) and 2% (Tier 2). Requires 104 Farming.

### Pontifex Observation Ring

The Pontifex observation ring acts as a Tier 1 luck item and improves the spawn rate of rockertunities.

## Methods

### Summary

See here to calculate how many ores will be obtained during training.

Levels Method Experience per hour
1-10 Copper/tin 5,000
10-20 Iron 15,000
20-30 Coal 25,000
30-40 Mithril 40,000
50-60 Runite 60,000
60-70 Orichalcite/Drakolith 70,000
70-80 Phasmatite/Necrite 85,000
80-90 Banite 110,000
89-97 Corrupted ore 180,000
90-97 Light/Dark animica 160,000
97-99+ Alaea crablets 190,000

### Levels 1-10: Copper/Tin

Players should begin by mining copper ore rocks and/or tin ore rocks. Mines containing these rocks include the Lumbridge Swamp training mining site, Burthorpe mine, and the South-east Varrock mining site. Players can continually mine from the same rock, but should switch to rocks with rockertunities when these spawn. With level 1-10 Mining players should gain an experience rate equivalent to around 5,000 experience an hour, and should achieve level 10 Mining after around 10 minutes.

### Levels 10-20: Iron

At level 10 players can start mining from iron ore rocks; these can be found in the South-west Lumbridge swamp mining site, South-west Varrock mining site, and Piscatoris mining site. Players should gain around 15,000 experience an hour at these levels. At level 15 players unlock the "energetic mining" mining mechanic which allows the player to deal increased damage per swing to rocks while mining, if they are actively paying attention. A player's activity level/stamina is indicated by coloured sparks around the rock being mined.

Players who are fully AFK gain no bonus, and will have no visible sparks. Those who maintain full stamina, by clicking a rock every ticks/2.4 seconds, will gain 25% additional damage per swing. This is indicated by green sparks. Players who are paying a moderate amount of attention, and maintaining between 1-99% stamina gain 12.5% additional damage, which is indicated by white sparks. This guide assumes that from level 15 onwards the player is maintaining a moderate level of activity; players who maintain 100% stamina may be able to gain high experience rates.

### Levels 20-30: Coal

From level 20-30 Mining players should mine coal rocks. The South-west Lumbridge swamp mining site, Falador mining site, and Barbarian Village mining site contain coal rocks. At these levels players should be able to gain around 25,000 experience an hour.

### Levels 30-40: Mithril

With level 30 Mining players can begin mining mithril ore rocks for around 40,000 experience an hour; however, it is ideal to start at level 31 for the doubled critical chance. Mithril ore rocks can be found at the South-east Varrock mining site, South-west Varrock mining site, and Karamja Volcano mine.

At level 40 players may chose to mine adamantite ore rocks or luminite rocks. However, it is ideal to start at level 45 for the doubled critical chance. Both ores give around 45,000 experience an hour. Luminite rocks are located at the Battlefield mining site, Dwarven Mine, Anachronia western ruin mine and the Mining Guild resource dungeon. Adamantite ore rocks can be found at the Rimmington mining site, Anachronia western ruin mine and South-east Varrock mining site.

### Levels 50-60: Runite

Runite ore rocks can be mined with at least level 50 Mining, however it is ideal to start at level 56 for the doubled critical chance. The rocks can be mined in the Mining Guild, Fight Arena mining site and South and North Crandor mining sites. This gives around 60,000 experience an hour.

### Levels 60-75: Orichalcite/Drakolith

From levels 60-70 Mining players may mine either orichalcite rocks or drakolith rocks; however, it is ideal to start at level 64 for the doubled critical chance. Orichalcite ore rocks may only be found in the Mining Guild, South-western Wilderness mining site and Anachronia canyon mine. Drakolith rocks can be found at the Karamja mining site, Heroes' Guild mine, Mining Guild resource dungeon, Wilderness level 14/15 just east of the wilderness lodgestone and Al Kharid mining site resource dungeon. By mining either type of rock the player will gain around 70,000 experience an hour.

### Levels 75-81: Phasmatite/Necrite

At level 70 players can mine either phasmatite rocks or necrite rocks. However, it is ideal to start at level 75 for the doubled critical chance, which gives around 85,000 - 95,000 experience an hour. Necrite rocks can be found in the Uzer mining site, Wilderness (LVL 39) Hobgoblin Mine and Al Kharid mining site resource dungeon; phasmatite rocks are located South of Port Phasmatys, in the Abandoned Mine and Dwarven Mine hidden mine (Hard Daemonheim tasks required).

### Levels 81-89: Banite

From level 80-90 players can mine banite ore for around 90,000-120,000 experience an hour; however, it is ideal to start at level 81 for the doubled critical chance. It can be mined from the Glacor Cave after Ritual of the Mahjarrat, from the Abomination Cave after Hero's Welcome, or from the Jatizso dungeon elite mine after completing the elite Fremennik achievements. Hallvar in the last location can note banite ore, which makes this the most convenient place to gather banite in large quantities. For players with a Dungeoneering skillcape, the Brimhaven metal dragons teleport can be used to quickly access the Abomination Cave mine. Without meeting any of these requirements, banite ore can be mined west of Trollweiss Mountain at the Arctic (azure) habitat mine, or at the Wilderness (Level 54) Pirates' Hideout mine west of the mage arena and east of the Wilderness Agility Course. The Pirates' Hideout mine is next to a metal bank, so this location is convenient for players willing to brave the wilderness.

### Levels 89-97: Corrupted ore

Starting at level 89 players who have completed Plague's End can mine corrupted ore for 130,000-200,000 experience an hour. Corrupted ore is mined from the Seren stones in the Trahaearn Clan's district of Prifddinas. Although corrupted ore is not a core rock and thus is ineligible for rockertunities or leveling bonuses, it has a 1.2 experience multiplier. This means that it will generally grant better experience per hour than banite or animica, which have 0.82 and 0.84 multipliers respectively. Corrupted ore also stacks, which makes it an excellent choice for AFK mining. Corrupted ore is untradeable, but can be smelted for fairly quick Smithing experience.

### Levels 90-97: Light/Dark animica

From level 90-97 players can mine either light or dark rocks. Players with level 96 Mining and the Elder rune pickaxe + 5 using Decorated mining urn (r) can expect between 110,000-120,000 experience per hour or around 90,000-110,000 experience per hour without urns. This jumps to around 160,000 experience per hour (with level 99 Mining and an elder rune pickaxe +5). However, it is ideal to start at level 91 for the doubled critical chance. Light animica rocks can be found on Anachronia South-West Mine, at Lletya mining site and at the entrance to the Smoke Dungeon; dark animica rocks are found on Anachronia Swamp Mine and within The Empty Throne Room. Additionally, during the Trahaearn Voice of Seren, rocks within the Trahaearn mines (excluding gem rocks) can randomly harmonize to a mixture of light and dark animica, potentially creating better arrangements for rockertunities, and giving 20% extra ores. Players can join the Friends Chat 'Harmed Ore' to find optimal arrangements of light or dark rocks during the Trahaearn Voice of Seren hour.

### Levels 97-99+: Alaea crablets

At level 97 Mining players can begin mining alaea crablets. Alaea crablets are a rare Mining resource that can be found randomly on Uncharted Isles. Uncharted Isles can be explored from The Arc region, which requires completion of the Impressing the Locals quest to access. Completion of the Flag Fall miniquest, which allows the player to claim an uncharted isle and return to it, is also recommended to hunt alaea crablets effectively. To find alaea crablets, the player must first gain some chimes from skilling in The Arc so that they can purchase the supplies required to voyage to Uncharted Isles. Supplies can be purchased from Rosie. When the player finds an Uncharted Isle inhabited by colonies of alaea crablets then they can claim the island using a claim island flag. Once an uncharted isle is claimed the player may return to it for the price of three supplies per voyage: five supplies per day can be claimed for free from the box next to Rosie. Up to three colonies of alaea crablets may be present on an Uncharted Isle, if the player opts to go on a long voyage.

Alaea crablets can be mined at level 97 Mining and yield alaea sea salt. Each day the player may mine for around 35 minutes at each colony of alaea crablets, gaining around 90,000 experience per colony before it is depleted. The daily cap resets at 00:00 UTC. When selling the alaea sea salt gained from mining alaea crablets players can expect to gain enough chimes to cover the cost of purchasing supplies to go on additional long voyages to find more alaea crablets and thus mine from them indefinitely. Purchasing the supplies cost reductions and commodity sell price increases unlocks from the Waiko Reward Shop, in addition to having a claimed island with as many alaea crablet colonies as possible, will further increase the viability of searching for crablets to mine continuously. At level 99 mining alaea crablets gives an experience rate equivalent to at around 190,000 Mining experience an hour.

The experience from mining alaea crablets can further be bolstered by completing contracts given by Sojobo on Waiko or via a contract contact on an uncharted isle. Players may complete up to three contracts daily, and may purchase upgrades from Boni for a maximum of seven daily contracts. Doing contracts, however, requires chimes to be able to re-roll contracts until satisfactory alaea crablet ones are assigned.

## Alternative methods of gaining experience

### Quests

Quest Members Experience
reward
Mining
requirement
Other requirements
The Tale of the Muspah 800 8
What's Mine is Yours 1,000 -
Legacy of Seergaze 2,000 35
Plague City 2,425 -
The Giant Dwarf 2,500 -
Heroes' Quest 2,575 50
Another Slice of H.A.M. 3,000 -
The Lost Tribe 3,000 17
Between a Rock... 5,000 40
Summer's End 5,000 45
A Void Dance 5,000 47
Enakhra's Lament 7,000 -
Defender of Varrock 10,000 59
Rebuilding Edgeville 10,000 -
Kennith's Concerns 12,000 46
TokTz-Ket-Dill 15,000 41
The Dig Site 15,300 -
The Curse of Arrav 30,000 64
Ritual of the Mahjarrat 40,000 76
King of the Dwarves 45,000 68
Plague's End 50,000 75
River of Blood 50,000 72
The Brink of Extinction 100,000 72
Birthright of the Dwarves 100,000 80
Total 589,600

### Wilderness Warbands

Wilderness Warbands is a dangerous Distraction and Diversion that takes place in the Wilderness. The aim of the Distraction and Diversion is to infiltrate a camp occupied by god-following NPCs and loot supplies, which can then be exchanged with Quercus for experience. Players can gain Mining experience if they obtain and hand in mining supplies. Players can gain between 49.5 at level 1, and 4,851.5 at level 99, Mining experience per supply that they exchange. Players can loot up to 25 mining supplies while playing Wilderness Warbands, which grant between 1,237.5 and 121,287.5 Mining experience in total when given to Quercus. Players can calculate the exact amount of experience that they will receive, based on their Mining level and the number of supplies obtained, using the calculator on the Wilderness Warbands article.

Wilderness Warbands takes place every seven hours. The Distraction and Diversion begins with a camp appearing in the Wilderness in one of three possible locations. A server-wide announcement will notify players of the camp's location when it spawns. Players can also find out the camp's location by talking to Quercus, who is located north-west of the Varrock Grand Exchange. Players must travel to the camp, avoid the sentries posted around its perimeter and sabotage the camp's god's beam, which is found in the centre of the camp. Successfully sabotaging the beam will result in NPCs allied to the invading players teleporting to the area to fight the NPCs guarding the camp. Players can also intervene in the fight if they wish. After all enemy NPCs have been killed, players can loot the tents in the camp to gain supplies. Each Wilderness Warband has three tents which contain supplies that grant experience in three of five possible skills (Mining, Farming, Construction, Herblore and Smithing). Players may loot up to 25 supplies from tents in the camp. Players may gain additional supplies by killing other players and taking any supplies that defeated players may have looted, as supplies are dropped upon death. Participating in looting tents will skull the player and cause them to become attackable by any other player, regardless of combat levels. Players cannot teleport out of the Wilderness while they have supplies in their inventory, and logging out while inside the Wilderness causes all supplies to be destroyed. Therefore, players must escape from the Wilderness by running out of the area. To avoid being killed by other players while participating in Wilderness Warbands players are advised to join one of the dedicated friends chats for the Distraction and Diversion.

Monday Tuesday Wednesday Thursday Friday Saturday Sunday
02:00 UTC 06:00 UTC 03:00 UTC 00:00 UTC 04:00 UTC 01:00 UTC 05:00 UTC
09:00 UTC 13:00 UTC 10:00 UTC 07:00 UTC 11:00 UTC 08:00 UTC 12:00 UTC
16:00 UTC 20:00 UTC 17:00 UTC 14:00 UTC 18:00 UTC 15:00 UTC 19:00 UTC
23:00 UTC N/A N/A 21:00 UTC N/A 22:00 UTC N/A

### Cabbage Facepunch Bonanza

Cabbage Facepunch Bonanza is an instanced minigame which can be played to gain renown points that can be exchanged for bonus experience in Mining. There are no requirements to play the minigame and players cannot bring any items into it. After learning how the game works and how to maximise points gained players can expect an average of 170 renown points per game depending on luck and other players. If a player is able to gain 170 renown per game, with each game lasting approximately 12 and a half minutes, 816 renown can be gained an hour. Each renown can be exchanged for between 2.2 with level 1, and 121.9, with level 99, Mining bonus experience; so players can gain between 1,795 and 99,470 bonus experience in Mining an hour. Each day players gain double renown for the first 600 renown that they obtain; therefore 1632 renown can be obtained in an hour, equivalent to between 3,590 and 198,941 bonus experience in Mining.

A game of Cabbage Facepunch Bonanza is composed of 10 rounds, during which players must repeatedly travel back and forth across an arena. Players have one minute and 10 seconds to cross the arena and enter the safe area each round. While crossing the arena players must avoid obstacles: wooden barricades block some routes through the arena and jets of fire appear intermittently in channels in the arena floor, dealing damage to any players who travel through them. gorilla guards and cabbage cauldrons also deal damage to players who come within proximity of them. An incremental amount of score is awarded to the player for each round that they survive i.e. players gain one score for surviving round 1, up to 10 score for surviving round 10. Players can gain an additional two score and have 10% of their maximum life points restored for each cabbage trophy that they find while in the arena. Cabbage trophies can be found by harvesting cabbage leaf patches and wishing-cabbage patches, mining cabbage vein ore and killing the weak, non-aggressive monkey minions. Multiple players can gain cabbage trophies and their health-restoring effects from the same source. At the end of the minigame the player is awarded an amount of renown which is equivalent to the score that they achieved during the game.

If a player does not reach the safe zone before the end of a round or loses all of their life points then they will be turned into a gorilla guard. As a gorilla guard, the player is severely limited in their ability to increase their score. Once per round a gorilla guard can lay a trap in the arena, which awards two scores. Gorillas can also attack non-gorilla players; all gorilla guards will receive two scores if a player is successfully killed. Gorilla guards do not take damage from obstacles in the arena.

Mining level Bonus experience granted per renown point
1 2.2
5
10 13.2
15 19.3
20 25.4
25 31.5
30 37.6
35 43.7
40 49.8
45 55.9
50 62.0
55 68.1
60 74.2
65 80.4
70 86.5
75 92.6
80 98.7
85 104.2
90
95 117.0
99 121.9

### Stealing Creation

Players can use reward points from the Stealing Creation minigame to purchase proto-tools and volatile clay tools. These items can be transformed into a selection of skill related tools, including sacred clay pickaxes and volatile clay pickaxes. These pickaxes can be right-clicked to grant Mining bonus experience (it is not necessary for the pickaxes to be wielded to gain bonus experience). Players are advised to play Stealing Creation within a group of players co-operating in non-combat games to deplete all sources of clay available more quickly. Once all the clay available in a game has been depleted the game will end early, allowing players to gain reward points in shorter amounts of time.

Both proto-tools and volatile clay tools cost 20 reward points per tool to purchase. When converting a proto-tool the player can choose which skilling tool it will become, but converting a volatile clay tool creates a random skilling tool. A volatile clay pickaxe grants 29,340 Mining bonus experience, while sacred clay pickaxes grant 24,450 Mining bonus experience. A skilling tool created from a volatile clay tool can be reverted back to the volatile clay tool and then converted again to potentially give a different tool. This may be done up to 10 times (the volatile clay tool is destroyed upon the 10th reversion); each reversion causing the tool formed from the volatile clay tool to have 10% less bonus experience available to be granted. Both sacred clay and volatile pickaxes retain their form after being drained of bonus experience, and players can then use 20 reward points to recharge the depleted tools rather than purchasing new tools. Because of this players can maximise the amount of Mining bonus experience that they obtain from their reward points by initially purchasing a volatile clay tool and then transforming and reverting the skilling tools created until a volatile clay pickaxe is obtained. Players can then continue recharging the volatile clay pickaxe, claiming 29,340 Mining bonus experience from it each time they obtain 20 reward points.

Players with at least level 80 in Mining, Hunter, Fishing and Woodcutting will be able to gather clay from all possible locations in a game of Stealing Creation. With these levels players can gather enough clay during a game of Stealing Creation to gain 24 reward points per game that they play. Assuming games of Stealing Creation are completed in around 12 and a half minutes, players can complete 4.8 games of Stealing Creation per hour and earn 115.2 reward points per hour - enabling them to obtain 140,832 Mining bonus experience an hour.

### Barbarian Assault

Players can gain Mining bonus experience as a reward from playing the Barbarian Assault minigame. At level 99 Mining it is possible to gain 1,900,000 bonus experience per hour playing hard mode Barbarian Assault waves 6-9, at a cost of six Barbarian assault ticket - hard wave 10 per hour. Players may as an alternative gain 1,300,000 bonus experience per hour playing hard mode waves 1-9 without having to pay for the tickets.

In order to play Hard mode Barbarian Assault the normal mode of the minigame must be completed at least once. It is highly recommended for players to reach maximum role levels before starting playing for bonus experience.

### Daeyalt ore

Players with a Mining level of 20 and during or after the quest The Darkness of Hallowvale can mine Daeyalt ore in the Meiyerditch mine. In order to reach the mine, players have to travel to Meiyerditch and get caught by one of the Vyrewatch, asking to be sent to the mines. The mines could also be accessed by navigating to the north-east corner of Meiyerditch.

There, players can mine Daeyalt ore rock for around 45,000-55,000 experience an hour. The ore is stackable, making this method completely AFK. Daeyalt rock is also a non-core rock, which generates no geodes, rockertunity or other mechanics that could fill the inventory or change its AFK nature. This leaves inventory space to other useful items such as Mining urns and Torstol incense sticks.

Players without River of Blood completed have to fill a mine cart with 15 Daeyalt ore to leave the mine.

### Menaphos worker district mining site

The Menaphos worker district mining site is located in the Worker district of Menaphos and contains 3 concentrated sandstone rocks, requiring level 50 Mining to mine. Players can gain up to 75,000 experience per hour mining sandstone rocks.

## Money-making using the Mining skill

Mining money making guides
Method Profit Skill Experience
Mining iron ore 949,000

10 (50 for Rune pickaxe)
15
25

48,960
Mining concentrated gold rocks 1,918,000 80
83 recommended
123,000
Mining adamantite ore 728,000 40 (50 recommended) 57,000
Mining runite ore 1,893,000 50
83
182,400
Mining runite ore 1,009,000 50
83
140,000
Mining runite ore 1,039,000 50 84,790
Mining clay 712,000 1 (20 for Steel pickaxe) 12,400
Mining soft clay 2,232,000 75 , , , , , , , and for access to Prifddinas. 70,000
Mining granite 2,024,000 45
68 recommended
66,540
Mining animica during Trahaearn Voice of Seren 1,293,000 90 (99 for Mining cape)
83
188,160
Mining luminite 1,067,000

40 (50 for Rune pickaxe)
15
25

70,000
Mining mithril ore 402,000

30 (50 for Rune pickaxe)
15
25

48,960
Mining copper ore 313,000 1 (20 for Steel pickaxe) 15,700
Mining tin ore 531,000 1 (20 for Steel pickaxe) 15700
Mining animica 1,553,000 90 (99 for Mining cape)
83
99 (optional for best rates)
117,600
Mining animica 1,602,000 90 (99 for Mining cape)
83
99 (optional for best rates)
117,600
Mining phasmatite ore 1,129,000 70 (76 - chance to get double ore)
83
99 (Optional for best rates)
80,000

### Recurring methods

Recurring Mining money making guides
Method Profit Time Effective Profit Recurrence time Experience
Mining red and crystal sandstone 1,521,000 20 4,562,000 24 hours 18750 , 10230


text:
## extending functions vs. extending objects

There's a talk by Prof Black about the origins of OO and Simula, and it make the point / the claim that it is easier to come up with objects/classes that are successfully malleable than it is to do so with functions; see pages 26 and 27 of the slides.

Parameterized functions instead of Inheritance? When you parameterize a function, you have to plan ahead and make parameters out of every part that could possibly change. functional programmers call this â€œabstractionâ€
Two problems: 1. Life is uncertain; 2. Most people think better about the concrete than the abstract.

The value of Inheritance When you inherit from a class or an object, you still have to plan ahead and make methods out of every part that could possibly change. o-o programmers call this â€œwriting short methodsâ€ Two benefits: 1. You donâ€™t have to get it right; 2. The short methods are concretions, not abstractions.

Given the problems of OO (in all its various guises), i've of late been leaning way more towards the FP side, so this was slightly interesting to read. :-)

## Comment viewing options

### Audrey Tang

I kinda like Audrey Tang's rule of thumb:

Things interacting in a small number of simple ways (anything)
Things interacting in a large number of simple ways (OO handles best)
Things interacting in a small number of complex ways (FP handles best)

Things interacting in a large number of complex ways what you need is a lot of small mostly independent programs loosely coupled. Generally a dynamic glue between the various pieces, and those pieces written as per the above.

In her original version it was VB for the interfaces, Haskell for engines and Perl for glue.

### Manycore

His point (though I think a bit too general) is a good one, "Most computing models treat computation as the expensive resource, it was so when those models were developed! Today: computation is free it happens â€œin the cracksâ€ between data fetch and data store Data movement is expensive both in time and energy."

His solution was pretty interesting, having controllers that allow for messages to be sent by descriptor or name:
i.e. Dec13th version of Oslo talk or

And then combining that with something like distributed version control... I'm not sure how you would handle the merge problem but the idea is fascinating.

### Quite bogus claim, but nice talk

This is a strange example of idea that, out of context, is somewhat convincing, but loses most of its ground when rephrased into the context of the talk.

Here is the argument of the speaker :
- Indeed, objects with inheritance can be encoded with functions and explicit open recursion
- but functions are less modular because you have to plan ahead parameter changes

This strikes me as a bogus claim. With the usual encoding of objects as a tuple/record of functions, each taking a "recursive" self parameter to model open recursion and dynamic dispatch, *you don't have to plan ahead*. You just need to add a new function/method somewhere, and call it through the self parameter in any place you wish. There is no plumbing as he claims because the plumbing is already done through self.

And this is perfectly natural, even obvious in retrospect : if you faithfully encode object plus inheritance in lambda-calculus, you get something which has just the same properties, with the same behavior wrt. code evolution, and also the same drawbacks wrt. difficult static reasoning due to the multitude of hard-to-specify extension points.

You could attack the idiomatic functional style to be somewhat different wrt. code evolution than the idiomatic object-oriented style. That's certainly true (new data vs. new operations, etc.; that's what JeffB is talking about in his reply). But attacking the functional translation of object-oriented style is bogus.

.

That said, I was rather pleased by the talk. There is a lot of historical research and the general idea that it's interesting to know about what people have done, in which context, etc. This is good. The technical content is quite fuzzy and wide-ranging, with suggestions rather than solutions, but could be considered thought-provoking.

There is a lot of whining about "complex time systems" in this talk that I felt somehow misdirected (all the more when you advertise programming style that do have more-complex-than-necessary behavior, bringing necessarily more complex static reasoning on the table). But I do not wish to start yet another static-vs-dynamic debate in this thread.

There is a funny idea that "object-oriented abstraction can exist without types" (around slide numbered 52 in the presentation and 92 in my PDF reader). He reuses the ADT (abstract data type) vs. Object distinction : one common but private implementation/representation of all ADTs element, vs. potentially different internals for objects of the same type/interface. His idea is that ADTs need a data representation (algebraic data type, concrete data) to benefit from the common representation, and therefore require type hiding to be effectively encapsulated. On the contrary, objects don't/can't/shouldn't expose their internals to each other, so can expose themselves as basically closures, black boxes, codata, that we aren't inspectable even in presence of type information. That's the usual remark that you can hide private state inside a closure (and you don't need existential types for that).

Before the final "concurrency, distribution, fault tolerance" pitch, there are a few statements that I found troubling.

Core Ideas of SIMULA
[...]
The actions and interactions of the objects created by the program model the actions and interactions of the real-world objects that they are designed to simulate.

I know that I have succeeded as a teacher when students anthropomorphize their objects

Another bit that is both weird and nice :

Algol 60 and Simula:
- Dahl recognized that the runtime structures of Algol 60 already contained the mechanisms that were necessary for simulation.
- It is the runtime behavior of a simulation system that models the real world, not the program text.

[...]

The Value of Dynamism:
Smalltalk is a â€œDynamic Languageâ€
Many features of the language and the programing environment help the programmer to interact with objects, rather than with code

Proposed definition: a â€œDynamic Programming Languageâ€ is one designed to facilitate the programmer learning from the run-time behavior of the program.

### Most people should

Most people should anthropomorphize their objects. Metaphorical thinking are incredibly useful in helping programmers get a handle on complexity. It can lead you in the wrong direction, of course (metaphors are always not perfect), but its a net plus rather than a net negative for most people. Exceptions are those that can think symbolically and logically, like Vulcans.

### Exceptions are those that

Exceptions are those that can think symbolically and logically, like Vulcans.

"...like intelligent humans" would be more accurate. Fiction like Star Trek often does a great job of stereotyping characteristics to turn them into easily discriminated-against packages, but it's particularly egregious in cases like this, where one of humanity's distinctive strengths is turned into a sort of slur.

### It wasn't meant as a slur.

It wasn't meant as a slur. There are people out there (for me, mostly in the FP crowd) who can think so well mathematically and symbolically that I can barely understand them. I like to think of Vulcan as a metaphor for those people; they seem like us but they can think in incredibly different ways. Perhaps they would find that kind of thinking offensive, since it is so metaphorical...

I think it is a fallacy to assume that (a) most programmers are like this or that (b) most programmers could be like this if only they studied/worked harder or saw "the light." I find that offensive, it is basically tantamount to cultural subjugation.

### Examples?

I disagree agree with this. The useful activity is model building. Programming is building a model and understanding how the problem domain maps to it. A good model is simple and a large catalog of elegant models is one of most important tools in a programmers toolbox.

So, to what extent are these models motivated by the real world? It seems to me most common that the real world analogy is found and used to explain the idea after the simple model is understood. For example, take "trees." I really doubt that one of our ancestors was looking at a tree and thought "that reminds me of relationships in our family" -- the real world concept of tree has too many irrelevant details. Rather, once the inheritance relation was drawn as a graph, someone thought to describe it as a tree. And while the tree analogy is useful for communication, as a tool for better understanding the model it probably doesn't ... bear much fruit.

So I'm curious, do you have any good examples of the analogy driving discovery? For me, looking back, it seems to usually be the models.

### How are our models not

How are our models not steeped in metaphors? It doesn't matter if the model is related to the real world or not, we will often apply metaphors to abstract concepts simply so that we can store and access them somewhere in our mind. A tree in programming only loosely resembles a natural tree, but that is irrelevant, we are just storing the abstraction somewhere in our mind where we can conveniently access the knowledge when we need it. For the same reason, we use our own spatial memory to reason about the state of a program, so using maps and dictionaries as metaphors is very useful to us even if the data structures only vaguely resemble their real life counterparts.

A pure model involves abstract symbols that mean little to us outside of the model (hence, they are abstract). This is quite useful when you want to analyze a system without bias, but during design its quite difficult for most of us to work with as we need metaphor and analogy to communicate it to others and evolve it forward. I'm not quite sure why this would be controversial here.

When you write a program do you think of it as a bunch of instructions evaluated by the computer or a living system with actors that do things? Of course, we have to translate it down to the computer, but I bet most of us do not start there.

### Ok

I apparently read too much into your use of "metaphor". I'm certainly not suggesting that programming is about algebraic manipulation or should be done in formal systems.


text:
# Ratio, Proportion and Rates of Change - Short Problems

This is part of our collection of Short Problems.

You may also be interested in our longer problems on Ratio, Proportion and Rates of Change.

Printable worksheets containing selections of these problems are available here.

### Three of a Kind

##### Age 11 to 14 Short Challenge Level:

How long is it between 5:55 and the next time all three digits on a digital clock are the same?

### Printer Ink

##### Age 11 to 14 Short Challenge Level:

A small ink cartridge has enough ink to print 600 pages. How many pages can be printed using a large ink cartridge?

### Flying Down Under

##### Age 11 to 14 Short Challenge Level:

Fred flew to Melbourne, Australia. What time was it in Melbourne when Fred's flight arrived?

### A Classy Ratio

##### Age 11 to 14 Short Challenge Level:

There are six more girls than boys in Miss Spelling's class of 24 pupils. What is the ratio of girls to boys in the class?

### Turbo Tortoise

##### Age 11 to 14 Short Challenge Level:

Harriet Hare and Turbo Tortoise want to cross the finish line together on their 12 mile race.... What time should Harriet set off?

### Brothers and Sisters

##### Age 11 to 14 Short Challenge Level:

Can you work out how many brothers and sisters are in the family?

### Dean's Mountain

##### Age 11 to 14 Short Challenge Level:

Dean runs down the mountain at 12 km per hour. How long does it take him to run down the mountain?

### Draining a Pool

##### Age 11 to 14 Short Challenge Level:

The water is being drained from a pool. After how long will the depth of the pool be 144 cm?

### Race Time

##### Age 11 to 14 Short Challenge Level:

Frank and Gabriel competed in a 200m race. Interpret the different units used for their times to work out who won.

### Athletics Club

##### Age 11 to 14 Short Challenge Level:

An athletics club has girl, boy and adult members. How many members does the club have?

### Minutes in Between

##### Age 11 to 14 Short Challenge Level:

How many minutes are there between 11:41 and 14:02?

### Speed over a Bridge

##### Age 11 to 14 Short Challenge Level:

How fast is the train travelling as it goes over the bridge?

### Family Fortune

##### Age 11 to 14 Short Challenge Level:

If three brothers will get £20 more if they do not share their money with their sister, how much money is there?

### Flapjack

##### Age 11 to 14 Short Challenge Level:

A recipe for flapjacks calls for a certain quantity of butter, sugar and oats. Given the amount of these ingredients I have, how many flapjacks can I make?

### Eating Beans

##### Age 11 to 14 Short Challenge Level:

The Bean family are very particular about beans. At every meal all Beans eat some beans... At their last meal they ate 23 beans altogether. How many beans did Pa Bean eat?

### 24 Hour Conundrum

##### Age 11 to 14 Short Challenge Level:

It is 225 minutes until midnight. What time is it on a 24-hour digital clock?

### Roman Distances

##### Age 11 to 14 Short Challenge Level:

Marcus' atrium was a square with each side 50 pedes long. How many times did Marcus have to walk round his atrium to complete his daily exercise of 8 stadia?

### Thunder and Lightning

##### Age 11 to 14 Short Challenge Level:

How far away was the lightning if the flash and the thunderclap were 6 seconds apart?

### Millennium Leap

##### Age 11 to 14 Short Challenge Level:

How many leap years will there be between 2001 and 3001?

### Decimal Clock

##### Age 11 to 14 Short Challenge Level:

A decimal clock is started at midnight. What time would it show at 6 o'clock in the morning?

### Off the Cuff

##### Age 11 to 14 Short Challenge Level:

Can you work out the ratio of shirt types made by a factory, if you know the ratio of button types used?

### Musical Maths

##### Age 11 to 14 Short Challenge Level:

Approximately how long is five hundred and twenty five thousand six hundred minutes?

### The Elephant Diet

##### Age 11 to 14 Short Challenge Level:

Use the relationship between the elephant and the rabbit to find out how many carrots the rabbit eats in a day

### Don't Be Late

##### Age 11 to 14 Short Challenge Level:

Mary is driving to Birmingham Airport. Using her average speed for the entire journey, find how long her journey took.

### Ratio Swap

##### Age 11 to 14 Short Challenge Level:

How many adults would need to join this group of people to reverse this ratio?

### Sharing Sweets

##### Age 11 to 14 Short Challenge Level:

Maria and Lucy spun a coin 30 times, and gave each other sweets depending on the outcome of the spin. How many times was tails spun?

### Tyre Wear

##### Age 11 to 14 Short Challenge Level:

A car with 5 tyres (four road tyres and a spare) travelled 30,000 km. All 5 tyres were used equally. How many kilometres' wear did each tyre receive?

### A Leg to Stand On

##### Age 11 to 14 Short Challenge Level:

Can you work out the number of chairs at a cafe from the number of legs?

##### Age 11 to 14 Short Challenge Level:

From two ratios of apples and oranges in a fruit bowl, can you work out how many fruits there are?

### How Many Swimmers?

##### Age 11 to 14 Short Challenge Level:

Given the ratios of junior to senior and senior to veteran members, can you work out what the smallest possible number of members in a swimming club is?

### Leg It

##### Age 11 to 14 Short Challenge Level:

How far from the finishing line should these runners start to make the race 'fair'?

### Out of Sync

##### Age 11 to 14 Short Challenge Level:

Albert Einstein could see two clocks which were out of sync. For what fraction of the day did they show the same time?

### Stair Climb

##### Age 11 to 14 Short Challenge Level:

Three people run up stairs at different rates. If they each start from a different point - who will win, come second and come last?

### Revolutions

##### Age 11 to 14 Short Challenge Level:

Jack and Jill run at different speeds in opposite directions around the maypole. How many times do they pass in the first minute?

### Washing Elephants

##### Age 11 to 14 Short Challenge Level:

How long will it take Mary and Nigel to wash an elephant if they work together?

### The London Eye

##### Age 14 to 16 Short Challenge Level:

The 80 spokes of The London Eye are made from 4 miles of cable. What is the approximate circumference of the wheel?

### Caught in the Web

##### Age 14 to 16 Short Challenge Level:

Can you estimate the mass of insects caught by spiders in the UK each year?

### Honey Bees

##### Age 14 to 16 Short Challenge Level:

How many bees could fly 1000 miles if they had 10 gallons of honey?

### Tummy Ache

##### Age 14 to 16 Short Challenge Level:

Timmy, Tammy and Tommy all go to the doctors. In what order do they arrive at the surgery?

### Hiking the Hill

##### Age 14 to 16 Short Challenge Level:

Sarah's average speed for a journey was 2 mph, and her return average speed was 4 mph. What is her average speed for the whole journey?

### Crude Calculation

##### Age 14 to 16 Short Challenge Level:

Louise has noticed that the price of petrol has changed. Can you work out the percentage that it has increased by?

### Walking in the Mountains

##### Age 14 to 16 Short Challenge Level:

Heidi and Peter pass two signs which say how far their destination is. How long will it take them to get there?

### Antifreeze

##### Age 14 to 16 Short Challenge Level:

How much more antifreeze is needed to make the proportion 30%?

### Inches and Barleycorns

##### Age 14 to 16 Short Challenge Level:

How many barleycorns are there in one inch?

### Travelling by Train

##### Age 14 to 16 Short Challenge Level:

Stephen stops at Darlington on his way to Durham. At what time does he arrive at Durham?

### Shepherd's Flock

##### Age 14 to 16 Short Challenge Level:

Does Joseph have too many sheep in his flock...

### Travelator

##### Age 14 to 16 Short Challenge Level:

When Andrew arrives at the end of the walkway, how far is he ahead of Bill?

### Hillwalking

##### Age 14 to 16 Short Challenge Level:

Andrew walks along a flat path, then up and down a hill, then back along the path. Is it possible to work out how far he has walked?

### Ratio Riddle

##### Age 14 to 16 Short Challenge Level:

Can you work out the ratio b:c given the ratios a:b and a:c?

### Very Long Line

##### Age 14 to 16 Short Challenge Level:

If the line on the right were 0.2mm thick, how long would it need to be to cover an area of one square metre?

### Overtake

##### Age 14 to 16 Short Challenge Level:

Tom and Tim are travelling towards Glasgow, but leave at different times. If Tim overtakes Tom, how fast is he travelling?

### Tyneside Average Speed

##### Age 14 to 16 Short Challenge Level:

Can you work out the average speed of the van?

### Platinum Puzzle

##### Age 14 to 16 Short Challenge Level:

What is the approximate total volume of platinum that has ever been produced?

### Pond Runners

##### Age 14 to 16 Short Challenge Level:

Can you work out how much time it will be before Katherine laps James around the pond?

### Bike Shop

##### Age 14 to 16 Short Challenge Level:

If I walk to the bike shop, but then cycle back, what is my average speed?

### Dolly Dolphin

##### Age 14 to 16 Short Challenge Level:

Can you find Dolly Dolphin's average speed as she swims with and against the current?

### Day Out

##### Age 14 to 16 Short Challenge Level:

Chris cycled faster than expected. Can you work out his average speed?

### Candles

##### Age 14 to 16 Short Challenge Level:

What is the ratio of the lengths of the candles?

### Petrol Stop

##### Age 14 to 16 Short Challenge Level:

From the information given, can you work out how long Roberto drove for after putting petrol in his car?

### Marathon Mission

##### Age 14 to 16 Short Challenge Level:

Minnie trained more for the London Marathon this year, so her speed increased. By what percentage did her time decrease?

### Walk up the Escalator

##### Age 14 to 16 Short Challenge Level:

Can you work out how long Aimee would take to get up the escalator if she walked?

### 100m Sprint

##### Age 14 to 16 Short Challenge Level:

Anna, Bridget and Carol run a race. Can you work out where Carol was when Anna finished?

### Flower Show

##### Age 14 to 16 Short Challenge Level:

How long will it take six gardeners to dig six circular flower beds?

### Boys and Girls

##### Age 14 to 16 Short Challenge Level:

Can you find the total number of students in the school, given some information about ratios?

### Televisual Technology

##### Age 14 to 16 Short Challenge Level:

If a new and an old television have screens of the same area, what is the ratio of their widths?

### Laps

##### Age 14 to 16 Short Challenge Level:

On which of the hare's laps will she first pass the tortoise?

### Two Trains

##### Age 14 to 16 Short Challenge Level:

Two trains started simultaneously, each travelling towards the other. How long did each train need to complete the journey?

### Fraction of Percentages

##### Age 14 to 16 Short Challenge Level:

What is $W$ as a fraction of $Z?$

### Traffic Jam

##### Age 14 to 16 Short Challenge Level:

Emily's journey home took 25% longer than usual. By what percentage was her average speed reduced?

### Late for Work

##### Age 14 to 16 Short Challenge Level:

What average speed should Ms Fanthorpe drive at to arrive at work on time?

### Backwards Laps

##### Age 14 to 16 Short Challenge Level:

If two friends run in opposite directions around a track, and they pass each other every 24 seconds, how long do they take to complete a lap?


text:
The physical fidelity and accuracy of the predicted grain growth depend on the accuracy of phase field parameters. In the Ginzburg-Landau equation

$$\frac{\partial \eta_i}{\partial t}=-L_i\left[\mu\left(\eta_i^3-\eta_i+2\gamma_i\eta_i\sum_{j\neq i}{\eta_j^2}\right)-\nabla\cdot\kappa_i\nabla\eta_i\right],$$

The parameters $$L_i$$$, $$\mu$$$, $$\gamma_i$$$and $$\kappa_i$$$ should be determined from GB properties (energy $$\sigma_{gb}$$$and mobility $$m_{gb}$$$), which are usually obtained from Molecular Dynamics computation. There are different methods around to relate these parameters with GB properties. The only requirement for the parametrization is that the parameterized model can reproduce the desired GB properties.

The phase field module implements the parameter determination method developed by Moelans et al. (Physical Review B, vol. 78, 024113, 2008). For isotropic GB properties, these phase field parameters are expressed as

$$\gamma = 1.5,$$

$$\kappa \approx \frac{3}{4} \sigma_{gb} l_{gb},$$

$$L \approx \frac{4}{3} \frac{m_{gb}}{l_{gb}},$$

$$\mu \approx \frac{3}{4} \frac{1}{f_{0,saddle}(\gamma)} \frac{\sigma_{gb}}{l_{gb}},$$

where $$l_{gb}$$$is the diffuse GB width, and $$f_{0,saddle}(\gamma)$$$ is the chemical free energy at the saddle point. The parameters $$(L,\kappa,\gamma,\mu)$$$are uniform across the system. The isotropic material model is implemented in GBEvolution. For systems with misorientation dependence of GB properties, a set of GB energies $$\sigma_{gb,k}$$$ and mobilities $$m_{gb,k}$$$as functions of misorientation angle $$k$$$ should be considered. An iterative procedure was used to calculate the parameters $$(L_k,\kappa_k,\gamma_k,\mu)$$$for each misorientation $$k$$$, which are then interpolated through the following functions to obtain the continuous phase field parameters at each integration point,

$$\psi=\frac{\sum\limits_{i=1}^{p}{\sum\limits_{j>i}^{p}{\psi_k\eta_i^2\eta_j^2}}}{\sum\limits_{i=1}^{p}{\sum\limits_{j>i}^{p}{\eta_i^2\eta_j^2}}},$$

where $$\psi_k$$$represents parameters $$L_k$$$, $$\kappa_k$$$, or $$\gamma_k$$$. $$\mu$$$is set to be constant. More details on the parametrization are referred to Moelans 2008. Double-circle grain test A double-circle grain test is used to validate the implementation of GB anisotropy with misorientation. In the right figure, two circular grains of the same initial size are embedded in a matrix grain, and they would shrink under the driving force of reducing total GB area. It can be proven that the changing grain area $$A$$$ of either circular grain should follow

$$A=A_0-2\pi m_{gb}\sigma_{gb}\cdot t,$$

where $$A_0$$$is the initial grain area. We performed two tests where different properties were assigned to GBs formed by the two circular grains and the matrix. First, these two GBs have the same boundary energy but different mobilities ($$m_2 = 2\cdot m_1$$$). Second, they obtain the same GB mobility but different energies ($$\sigma_2 = 2 \cdot \sigma_1$$$). The simulated and analytical grain area evolution are compared in following figures. Area evolution of Grain 1 Area evolution of Grain 2 Apart from little non-linearity at the beginning of the 2$$^{nd}$$$ test simulation due to GB relaxation from symmetrical to non-symmetrical profile, the grain area evolutions agree quite well with analytical solution, which shows the phase field parameters determined from this method reproduces exactly the desired GB properties.

The anisotropic GB properties are implemented in material GBAnisotropy to determine the phase field parameters.