The Sign of One

Make the following numbers using FOUR 1s using any mathematics operators and/or symbols, such as $\dfrac{x}{y}$, $\sqrt{x}$, decimal dots, $+$, $-$, $\times$, $\div$, $($ $)$, etc by the Sign of One.

Click the numbers below to see the answers.

The first one is done for you.

$1$

$1 = 1 \times 1 \times 1 \times 1$

$2$

$2 = (1+1) \times 1 \times 1$

$3$

$3 = (1+1+1) \times 1$

$4$

$4 = 1+1+1+1$

$5$

$5 = \dfrac{1}{.1} \div (1+1)$
Note that the dot is the decimal point, such as $.1 = 0.1$

$6$

\begin{align} \displaystyle 6 &= (1+1+1)! \times 1 \\ &\text{Note that the exclamation mark ! means factorial, such as } 3! = 3 \times 2 \times 1 = 24 \\ &= \sqrt{\dfrac{1}{.\dot{1}}} + \sqrt{\dfrac{1}{.\dot{1}}} \\ &\text{Note that } .\dot{1} = .111 \cdots \text{ recurring decimals.} \\ &= \Bigg(\sqrt{\dfrac{1}{.\dot{1}}}\Bigg)! \\ \end{align}

$7$

$7 = (1+1+1)! + 1$

$8$

$8 = \dfrac{1}{.\dot{1}} – 1 \times 1$

$9$

$9 = \dfrac{1}{.\dot{1}} + 1 – 1$

$10$

$10 = \dfrac{1}{.\dot{1}} + 1 \times 1$

$11$

$11 = \dfrac{1}{.\dot{1}} + 1 + 1$

$12$

$12 = 11 + 1 \times 1$

$13$

$13 = 11 + 1 + 1$

$14$

$14 = 11 + \sqrt{\dfrac{1}{.\dot{1}}}$

$15$

$15 = \dfrac{1}{.\dot{1}} + \Bigg(\sqrt{\dfrac{1}{.\dot{1}}}\Bigg)!$

$16$

$16 = \dfrac{1}{.1} + \Bigg(\sqrt{\dfrac{1}{.\dot{1}}}\Bigg)!$

$17$

$17 = 11 + \Bigg(\sqrt{\dfrac{1}{.\dot{1}}}\Bigg)!$

$18$

$18 = \dfrac{1}{.\dot{1}} + \dfrac{1}{.\dot{1}}$

$19$

$19 = \dfrac{1}{.1} + \dfrac{1}{.\dot{1}}$

$20$

$20 = \dfrac{1}{.1} + \dfrac{1}{.1}$

$21$

$21 = \dfrac{1}{.1} + 11$

$22$

$22 = 11 + 11$

$23$

$23 = \Bigg(\sqrt{\dfrac{1}{.\dot{1}}}+1\Bigg)!-1$

$24$

$24 = \Bigg(\sqrt{\dfrac{1}{.\dot{1}}}+1\Bigg)!\times 1$

$25$

$25 = \Bigg(\sqrt{\dfrac{1}{.\dot{1}}}+1\Bigg)!+1$

$26$

\begin{align} \displaystyle 26 &= \Biggl\lceil \sqrt{\dfrac{1}{.1}} \Biggl\rceil ! + 1 + 1 \\ &= \Bigl\lceil \sqrt{10} \Bigl\rceil ! + 2 \\ &= \bigl\lceil 3.162 \cdots \bigr\rceil ! + 2 \\ &= 4! + 2 \\ &= 4 \times 3 \times 2 \times 1 + 2 \\ &= 24 + 2 \\ &= 26 \text{ Boom!}\\ \text{Note that:} \\ \bigl\lfloor x \bigr\rfloor &= \text{the greatest integer less than or equal to } x \text{ (floor)} \\ \bigl\lfloor 3.4 \bigr\rfloor &= 3 \\ \bigl\lceil x \bigr\rceil &= \text{the smallest integer greater than or equal to } x \text{ (ceiling)} \\ \bigl\lceil 3.4 \bigr\rceil &= 4 \\ \end{align}

$27$

\begin{align} \displaystyle 27 &= \Biggl\lceil \sqrt{\dfrac{1}{.1}} \Biggl\rceil ! + \Biggl\lfloor \sqrt{\dfrac{1}{.1}} \Biggl\rfloor \\ &= \Bigl\lceil \sqrt{10} \Bigl\rceil ! + \Bigl\lfloor \sqrt{10} \Bigl\rfloor \\ &= \bigl\lceil 3.162 \cdots \bigl\rceil ! + \bigl\lfloor 3.162 \cdots \bigl\rfloor \\ &= 4! + 3 \\ &= 4 \times 3 \times 2 \times 1 + 3 \\ &= 24 + 3 \\ &= 27 \\ \end{align}

$28$

\begin{align} \displaystyle 28 &= \Biggl\lceil \sqrt{\dfrac{1}{.1}} \Biggl\rceil ! + \Biggl\lceil \sqrt{\dfrac{1}{.1}} \Biggl\rceil \\ &= \bigl\lceil 3.162 \cdots \bigl\rceil ! + \bigl\lceil 3.162 \cdots \bigl\rceil \\ &= 4! + 4 \\ &= 4 \times 3 \times 2 \times 1 + 4 \\ &= 24 + 4 \\ &= 28 \\ \end{align}