# four fours problem

The four fours problem is as follows: using arithmetic combinations of four 4's express all the numbers from 1
to 100. For example, 1 = 44/44 and 2 = (4 × 4)/(4 + 4). The problem
was first presented in *The Schoolmaster's Assistant: Being a Compendium
of Arithmetic Both Practical and Theoretical* (first edition c. 1744),
a popular textbook by the English schoolteacher and cleric Thomas Dilworth
(d. 1780).

Operations and symbols that are allowed include the four arithmetic operations
(+, × , –, /), concatenation (e.g. the use of 44), decimal points (e.g.
4.4), powers (e.g. 44), square roots, factorials (4!), and overbars for
repeating digits (e.g. .4 with an overbar to express 4/9). Ordinary use
of parentheses are allowed. One of the trickiest numbers to represent in
this way is 73, which calls for something as a contorted as √(√(√(44!)))
+ 4 / .4' (where .4' is shorthand for .444...). Of course, the problem can
be extended to represent integers greater than 100. The highest value achievable
in the four four's puzzle is 10^{8.0723047260281×10^153} =
4^{4^4^4}.

S. K. Johnson has pointed out that a higher value, using the factorial,
is 4!^{4!^4!^4!}. Thanks to Amory Wong for pointing out that an
even larger value is 4444!