Four fours

Four fours is a mathematical puzzle. The goal of four fours is to find the simplest mathematical expression for every whole number from 0 to some maximum, using only common mathematical symbols and the digit four (no other digit is allowed). Most versions of four fours require that each expression have exactly four fours, but some variations require that each expression have the minimum number of fours.

The first printed occurrence of this activity is in "Mathematical Recreations and Essays" by W. W. Rouse Ball published in 1892. In this book it is described as a "traditional recreation".

In his discussion of the problem Ball calls it "An arithmetical amusement, said to have been first propounded in 1881, ...." . This date aligns with the appearance of the problem in Knowledge: An Illustrated Magazine of Science, (Dec 30, 1881) edited by Richard A. Proctor, the English astronomer who is remembered for one of the earliest maps of Mars.

There are many variations of four fours; their primary difference is which mathematical symbols are allowed. Essentially all variations at least allow addition ("+"), subtraction ("−"), multiplication ("×"), division ("÷"), and parentheses, as well as concatenation (e.g., "44" is allowed). Most also allow the factorial ("!"), exponentiation (e.g. "44⁴"), the decimal point (".") and the square root ("√") operation. Other operations allowed by some variations include the reciprocal function ("1/x"), subfactorial ("!" before the number: !4 equals 9), overline (an infinitely repeated digit), an arbitrary root, the gamma function (Γ(), where Γ(x) = (x − 1)!), and percent ("%"). Thus 4/4% = 100 and Γ(4)=6. A common use of the overline in this problem is for this value:

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