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 \( .\dot{1} = 0.111 \cdots \)
\( \displaystyle 9 = \frac{1}{.\dot{1}} +1-1 \)
\( \displaystyle 10 = \frac{1}{.\dot{1}} + 1 \times 1 \)
\( \displaystyle 11 = \frac{1}{.\dot{1}} + 1 + 1 \)
\( 12 = 11 + 1 \times 1 \)
\( 13 = 11 + 1 + 1 \)
\( \displaystyle 14 = 11 + \sqrt{\frac{1}{.\dot{1}}} \)
\( \displaystyle 15 = \frac{1}{.\dot{1}} + \left( \sqrt{\frac{1}{.\dot{1}}} \right)! \)
\( \displaystyle 16 = \frac{1}{.1} + \left( \sqrt{\frac{1}{.\dot{1}}} \right)! \)
\( \displaystyle 17 = 11 + \Bigg(\sqrt{\dfrac{1}{.\dot{1}}}\Bigg)! \)
\( \displaystyle 18 = \dfrac{1}{.\dot{1}} + \dfrac{1}{.\dot{1}} \)
\( \displaystyle 19 = \dfrac{1}{.1} + \dfrac{1}{.\dot{1}} \)
\( \displaystyle 20 = \dfrac{1}{.1} + \dfrac{1}{.1} \)
\( \displaystyle 21 = \dfrac{1}{.1} + 11 \)
\( \displaystyle 22 = 11 + 11 \)
\( \displaystyle 23 = \Bigg(\sqrt{\dfrac{1}{.\dot{1}}}+1\Bigg)!-1 \)
\( \displaystyle 24 = \Bigg(\sqrt{\dfrac{1}{.\dot{1}}}+1\Bigg)!\times 1 \)
\( \displaystyle 25 = \Bigg(\sqrt{\dfrac{1}{.\dot{1}}}+1\Bigg)!+1 \)
\( \begin{align} \displaystyle
26 &= \Biggl\lceil \sqrt{\dfrac{1}{.1}} \Biggl\rceil ! + 1 + 1 \\
&= \Bigl\lceil \sqrt{10} \Bigl\rceil ! + 2 \\
&= \bigl\lceil 3.162 \cdots \bigr\rceil ! + 2 \\
&= 4! + 2 \\
&= 4 \times 3 \times 2 \times 1 + 2 \\
&= 24 + 2 \\
&= 26 \text{ Boom!}\\
\text{Note that:} \\
\bigl\lfloor x \bigr\rfloor &= \text{the greatest integer less than or equal to } x \text{ (floor)} \\
\bigl\lfloor 3.4 \bigr\rfloor &= 3 \\
\bigl\lceil x \bigr\rceil &= \text{the smallest integer greater than or equal to } x \text{ (ceiling)} \\
\bigl\lceil 3.4 \bigr\rceil &= 4 \\
\end{align} \)
\( \begin{align} \displaystyle
27 &= \Biggl\lceil \sqrt{\dfrac{1}{.1}} \Biggl\rceil ! + \Biggl\lfloor \sqrt{\dfrac{1}{.1}} \Biggl\rfloor \\
&= \Bigl\lceil \sqrt{10} \Bigl\rceil ! + \Bigl\lfloor \sqrt{10} \Bigl\rfloor \\
&= \bigl\lceil 3.162 \cdots \bigl\rceil ! + \bigl\lfloor 3.162 \cdots \bigl\rfloor \\
&= 4! + 3 \\
&= 4 \times 3 \times 2 \times 1 + 3 \\
&= 24 + 3 \\
&= 27 \\
\end{align} \)
\( \begin{align} \displaystyle
28 &= \Biggl\lceil \sqrt{\dfrac{1}{.1}} \Biggl\rceil ! + \Biggl\lceil \sqrt{\dfrac{1}{.1}} \Biggl\rceil \\
&= \bigl\lceil 3.162 \cdots \bigl\rceil ! + \bigl\lceil 3.162 \cdots \bigl\rceil \\
&= 4! + 4 \\
&= 4 \times 3 \times 2 \times 1 + 4 \\
&= 24 + 4 \\
&= 28 \\
\end{align} \)
Responses