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.
$1$
$2$
$3$
$4$
$5$
$5 = \dfrac{1}{.1} \div (1+1)$
Note that the dot is the decimal point, such as $.1 = 0.1$
$6$
\( \begin{align} \displaystyle
6 &= (1+1+1)! \times 1 \\
&\text{Note that the exclamation mark ! means factorial, such as } 3! = 3 \times 2 \times 1 = 24 \\
&= \sqrt{\dfrac{1}{.\dot{1}}} + \sqrt{\dfrac{1}{.\dot{1}}} \\
&\text{Note that } .\dot{1} = .111 \cdots \text{ recurring decimals.} \\
&= \Bigg(\sqrt{\dfrac{1}{.\dot{1}}}\Bigg)! \\
\end{align} \)
$7$
$8$
$9$
$10$
$11$
$12$
$13$
$14$
$15$
$16$
$17$
$18$
$19$
$20$
$21$
$22$
$23$
$24$
$25$
$26$
\( \begin{align} \displaystyle
26 &= \Biggl\lceil \sqrt{\dfrac{1}{.1}} \Biggl\rceil ! + 1 + 1 \\
&= \Bigl\lceil \sqrt{10} \Bigl\rceil ! + 2 \\
&= \bigl\lceil 3.162 \cdots \bigr\rceil ! + 2 \\
&= 4! + 2 \\
&= 4 \times 3 \times 2 \times 1 + 2 \\
&= 24 + 2 \\
&= 26 \text{ Boom!}\\
\text{Note that:} \\
\bigl\lfloor x \bigr\rfloor &= \text{the greatest integer less than or equal to } x \text{ (floor)} \\
\bigl\lfloor 3.4 \bigr\rfloor &= 3 \\
\bigl\lceil x \bigr\rceil &= \text{the smallest integer greater than or equal to } x \text{ (ceiling)} \\
\bigl\lceil 3.4 \bigr\rceil &= 4 \\
\end{align} \)
$27$
\( \begin{align} \displaystyle
27 &= \Biggl\lceil \sqrt{\dfrac{1}{.1}} \Biggl\rceil ! + \Biggl\lfloor \sqrt{\dfrac{1}{.1}} \Biggl\rfloor \\
&= \Bigl\lceil \sqrt{10} \Bigl\rceil ! + \Bigl\lfloor \sqrt{10} \Bigl\rfloor \\
&= \bigl\lceil 3.162 \cdots \bigl\rceil ! + \bigl\lfloor 3.162 \cdots \bigl\rfloor \\
&= 4! + 3 \\
&= 4 \times 3 \times 2 \times 1 + 3 \\
&= 24 + 3 \\
&= 27 \\
\end{align} \)
$28$
\( \begin{align} \displaystyle
28 &= \Biggl\lceil \sqrt{\dfrac{1}{.1}} \Biggl\rceil ! + \Biggl\lceil \sqrt{\dfrac{1}{.1}} \Biggl\rceil \\
&= \bigl\lceil 3.162 \cdots \bigl\rceil ! + \bigl\lceil 3.162 \cdots \bigl\rceil \\
&= 4! + 4 \\
&= 4 \times 3 \times 2 \times 1 + 4 \\
&= 24 + 4 \\
&= 28 \\
\end{align} \)
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