變成 6

把數字湊成6。

2023/08/23

加上一些符號，讓下列的式子成立

0 \quad 0 \quad 0 = 6 \\ 1 \quad 1 \quad 1 = 6 \\ 2 \quad 2 \quad 2 = 6 \\ 3 \quad 3 \quad 3 = 6 \\ 4 \quad 4 \quad 4 = 6 \\ 5 \quad 5 \quad 5 = 6 \\ 6 \quad 6 \quad 6 = 6 \\ 7 \quad 7 \quad 7 = 6 \\ 8 \quad 8 \quad 8 = 6 \\ 9 \quad 9 \quad 9 = 6

例如

2 + 2 + 2 = 6 \\ 3 \times 3 - 3 = 6

參考解答

\begin{align*} (0! + 0! + 0! ) ! & = 6 \\ (1 + 1 + 1) ! & = 6 \\ 2 + 2 + 2 & = 6 \\ 3 \times 3 - 3 & = 6 \\ 4 + \sqrt{\sqrt{4 \times 4}} & = 6 \\ 5 + (5 \div 5) & = 6 \\ 6 + 6 - 6 & = 6 \\ 7 - (7 \div 7) & = 6 \\ 8 - \sqrt{\sqrt{8 + 8}} & = 6 \\ (9 + 9) \div \sqrt{9} & = 6 \end{align*}

這邊一共用了 $+$ , $-$ , $\times$ , $\div$ , $!$ , $()$ , $\sqrt{}$ 共7種操作，能不能只用到5種操作？

\begin{align*} (0! + 0! + 0!)! & = 6 \\ (1 + 1 + 1)! & = 6 \\ \left( \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{2} \rfloor \right)! & = 6 \\ \left( \lfloor \sqrt{3} \rfloor + \lfloor \sqrt{3} \rfloor + \lfloor \sqrt{3} \rfloor \right)! & = 6 \\ \left( \left\lfloor \sqrt{\sqrt{4}} \right\rfloor + \left\lfloor \sqrt{\sqrt{4}} \right\rfloor + \left\lfloor \sqrt{\sqrt{4}} \right\rfloor \right)! & = 6 \\ \left( \left\lfloor \sqrt{\sqrt{5}} \right\rfloor + \left\lfloor \sqrt{\sqrt{5}} \right\rfloor + \left\lfloor \sqrt{\sqrt{5}} \right\rfloor \right)! & = 6 \\ \left( \left\lfloor \sqrt{\sqrt{6}} \right\rfloor + \left\lfloor \sqrt{\sqrt{6}} \right\rfloor + \left\lfloor \sqrt{\sqrt{6}} \right\rfloor \right)! & = 6 \\ \left( \left\lfloor \sqrt{\sqrt{7}} \right\rfloor + \left\lfloor \sqrt{\sqrt{7}} \right\rfloor + \left\lfloor \sqrt{\sqrt{7}} \right\rfloor \right)! & = 6 \\ \left( \left\lfloor \sqrt{\sqrt{8}} \right\rfloor + \left\lfloor \sqrt{\sqrt{8}} \right\rfloor + \left\lfloor \sqrt{\sqrt{8}} \right\rfloor \right)! & = 6 \\ \left( \left\lfloor \sqrt{\sqrt{9}} \right\rfloor + \left\lfloor \sqrt{\sqrt{9}} \right\rfloor + \left\lfloor \sqrt{\sqrt{9}} \right\rfloor \right)! & = 6 \end{align*}

或甚至更簡潔的操作

\begin{align*} (0! + 0! + 0!)! & = 6 \\ \left( \left( \frac{d}{dx} 1 \right) ! + \left( \frac{d}{dx} 1 \right) ! + \left( \frac{d}{dx} 1 \right) ! \right) ! & = 6 \\ \left( \left( \frac{d}{dx} 2 \right) ! + \left( \frac{d}{dx} 2 \right) ! + \left( \frac{d}{dx} 2 \right) ! \right) ! & = 6 \\ \left( \left( \frac{d}{dx} 3 \right) ! + \left( \frac{d}{dx} 3 \right) ! + \left( \frac{d}{dx} 3 \right) ! \right) ! & = 6 \\ \left( \left( \frac{d}{dx} 4 \right) ! + \left( \frac{d}{dx} 4 \right) ! + \left( \frac{d}{dx} 4 \right) ! \right) ! & = 6 \\ \left( \left( \frac{d}{dx} 5 \right) ! + \left( \frac{d}{dx} 5 \right) ! + \left( \frac{d}{dx} 5 \right) ! \right) ! & = 6 \\ \left( \left( \frac{d}{dx} 6 \right) ! + \left( \frac{d}{dx} 6 \right) ! + \left( \frac{d}{dx} 6 \right) ! \right) ! & = 6 \\ \left( \left( \frac{d}{dx} 7 \right) ! + \left( \frac{d}{dx} 7 \right) ! + \left( \frac{d}{dx} 7 \right) ! \right) ! & = 6 \\ \left( \left( \frac{d}{dx} 8 \right) ! + \left( \frac{d}{dx} 8 \right) ! + \left( \frac{d}{dx} 8 \right) ! \right) ! & = 6 \\ \left( \left( \frac{d}{dx} 9 \right) ! + \left( \frac{d}{dx} 9 \right) ! + \left( \frac{d}{dx} 9 \right) ! \right) ! & = 6 \end{align*}