# 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.

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

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

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

$4 = 1+1+1+1$

$\displaystyle 5 = \frac{1}{.1} ] \div (1+1)$

Note that the dot ius the decimal point such as $.1 = 0.1$

$(1+1+1)! \times 1$

Note that the exclamation mark ! means factorial such as $3! = 3 \times 2 \times 1 = 6$

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

$\displaystyle 8 = \frac{1}{.\dot{1}}-1 \times 1$

Note that $.\dot{1} = 0.111 \cdots$

$\displaystyle 9 = \frac{1}{.\dot{1}} +1-1$

$\displaystyle 10 = \frac{1}{.\dot{1}} + 1 \times 1$

$\displaystyle 11 = \frac{1}{.\dot{1}} + 1 + 1$

$12 = 11 + 1 \times 1$

$13 = 11 + 1 + 1$

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

$\displaystyle 15 = \frac{1}{.\dot{1}} + \left( \sqrt{\frac{1}{.\dot{1}}} \right)!$

$\displaystyle 16 = \frac{1}{.1} + \left( \sqrt{\frac{1}{.\dot{1}}} \right)!$

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

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

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

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

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

$\displaystyle 22 = 11 + 11$

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

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

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

\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}

\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}

\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}

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