Four fours is a mathematical puzzle that asks for representations of positive integers using exactly four occurrences of the digit 4 and a specified collection of operations. A traditional version permits addition, subtraction, multiplication, division, parentheses, square roots, factorials, decimal points, and concatenation, as in 44 (Ball 1914, p. 14; Numberphile 2017).
The permitted operations materially affect the problem. If natural logarithms are admitted, every nonnegative
integer has a four-fours representation. Let 
denote 4 with 
nested square roots, so 
for the nonnegative
integer 
.
Then
 |
After 
is expanded, the right side contains
exactly four occurrences of the digit 4. The identity
follows from 
and 
. It uses arbitrarily many
square roots as 
increases.
