Four Fours/Lemmata/Two Fours/64/Solutions/5
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Puzzle: Two Fours: $64$
Using exactly $2$ instances of the number $4$, the task is to write an expression for $64$, using whatever arithmetical operations you consider necessary.
Solution
- $64 = \paren {\map \phi {4!} }^\sqrt 4$
where $\phi$ denotes the Euler $\phi$ function.
Proof
| \(\ds \paren {\map \phi {4!} }^\sqrt 4\) | \(=\) | \(\ds \paren {\map \phi {24} }^2\) | Definition of Factorial, Definition of Square Root | |||||||||||
| \(\ds \) | \(=\) | \(\ds 8^2\) | Euler Phi Function of 24 | |||||||||||
| \(\ds \) | \(=\) | \(\ds 64\) |
$\blacksquare$
Sources
- 1964: Donald E. Knuth: Representing Numbers using Only One Four (Math. Mag. Vol. 37: pp. 308 – 310) www.jstor.org/stable/2689238