Four Fours/Lemmata/Two Fours/64/Solutions/2

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

$64 = \sqrt {\paren {\sqrt {\sqrt 4} }^{4!} }$

Proof

$\sqrt {\paren {\sqrt {\sqrt 4} }^{4!} } = \sqrt {\sqrt {2^{24} } } = \sqrt {2^{12} } = \sqrt {4096} = 64$

Also presented as

Henry Ernest Dudeney's presentation of this (of which the above is a reproduction) is not the neatest.

This is how it appears according to Donald Ervin Knuth:

$64 = \surd \surd \surd \paren {4^{4!} }$

Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $58$. -- The Two Fours
• 1964: Donald E. Knuth: Representing Numbers using Only One Four (Math. Mag. Vol. 37: pp. 308 – 310)  www.jstor.org/stable/2689238
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $109$. The Two Fours