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 108.0723047260281×10^153 = 44^4^4.


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!