Henry Ernest Dudeney/Modern Puzzles/58 - The Two Fours

by : $58$

 * The Two Fours
 * The point [of the Four Fours puzzle] is to express all possible whole numbers with four fours (no more and no fewer), using the various arithmetical signs.


 * Thus $4 \times 4 + \tfrac 4 4$ equals $17$, and $44 + 4 + \sqrt 4$ equals $50$.


 * All numbers up to $112$ inclusive may be solved, using only the signs for addition, subtraction, multiplication, division, square root, decimal points, and the factorial sign $4!$ which means $1 \times 2 \times 3 \times 4$, or $24$, but $113$ is impossible.


 * It is necessary to discover which numbers can be formed with one four, with two fours, and with three fours, and to record these for combination as required.


 * It is the failure to find some of these that leads to so much difficulty.


 * For example, I think very few discover that $64$ can be expressed with only two fours.


 * Can the reader do it?