## Pairs and squares

This puzzle, which I posted to twitter (@puzzlecritic) a few weeks ago, is one of my variations on a problem that appeared in a mathematical discussion group on LinkedIn:

Find three different positive integers such that the sum of any two is a square.

## The Vault: Powers of two

My new book of maths puzzles is on its way! It’s packed full of interesting problems to sink your teeth into. I’ll post an update as the launch approaches.

In the meantime, here is a fantastic problem from the USA:

It is given that $2^{2004}$ is a 604-digit number beginning with a 1. How many of the numbers $2^0, 2^1, 2^2, 2^3, ..., 2^{2003}$ begin with a 4?

## Shares

From the 2004 Tournament of Towns:

Each day, the price of the shares of the corporation “Soap Bubble, Limited” either increases or decreases by n%, where n is an integer such that 0<n<100. The price is calculated with unlimited precision. Does there exist an n for which the price can take the same value twice?

## 42 Points

From the 2007 Australian Maths Competition:

There are 42 points $P_1, P_2, ..., P_{42}$ , placed in order on a straight line so that each distance from $P_i$ to $P_{i+1}$ is  $\dfrac{1}{i}$  for $1\leq i\leq 41$. What is the sum of the distances between every pair of these points?

## Rugs

From the 1998 University of Waterloo Fermat Contest:

Three rugs have a combined area of 200 sq m. By overlapping rugs to cover a floor of area 140 sq m, the area which is covered by exactly two layers of rug is 24 sq m. What area of floor is covered by three layers of rug?

## A curious operation

From the Putnam Competition (I believe):

Suppose # is a binary operation on a set S such that the following properties hold:

1. For all a, b, c in S, (a#b)#c = a#(b#c);
2. For all a, b in S, if a#b = b#a then a=b.

Prove that, for all x, y, z in S, we have x#y#z = x#z.