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Four fours

Four fours is a mathematical game. It is often used with older children to explore numbers and mathematical expressions, but many adults have also found it enjoyable.

The goal of four fours is to find a 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.

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 ("*" in ASCII), division ("/"), and parentheses, as well as concatenation (e.g., "44" is allowed). Most also allow the square root operation, factorial ("!"), exponentiation ("^" in ASCII), and the decimal digit ("."). Other operations allowed by some variations include overline (an infinitely repeated digit), an arbitrary root power, the gamma function (Γ(), where Γ(x) = (x-1)!), and percent ("%"). Typically the "log" operators are not allowed, since there's a way to trivially create any number using them. Paul Bourke credits Ben Rudiak-Gould with this description of how natural logarithms (ln()) can be used to represent any positive integer n as: <math>n = -ln[ ln( sqrt(sqrt(...(sqrt(4))...))) / ln(4) ] / ln(4)</math> where the number of nested sqrt() functions is twice n.

Additional variants (usually no longer called "four fours") replace the set of digits ("4, 4, 4, 4") with some other set of digits, say of the birthyear of someone. For example, a variant using "1975" would require each expression to use one 1, one 9, one 7, and one 5.

Here is a set of four fours solutions for the numbers 0 through 20, using typical rules:

  • 0 = 44-44
  • 1 = 44/44
  • 2 = 4/4+4/4
  • 3 = (4+4+4)/4
  • 4 = 4*(4-4)+4
  • 5 = (4*4+4)/4
  • 6 = 4*.4+4.4
  • 7 = 44/4-4
  • 8 = 4+4.4-.4
  • 9 = 4+4+4/4
  • 10 = 44/4.4
  • 11 = 4/.4+4/4
  • 12 = (44+4)/4
  • 13 = 4!-44/4
  • 14 = 4*(4-.4)-.4
  • 15 = 44/4+4
  • 16 = .4*(44-4)
  • 17 = 4*4+4/4
  • 18 = 44*.4+.4
  • 19 = 4!-4-4/4
  • 20 = 4*(4/4+4)

Certain numbers, such as 113 and 123, are particularly difficult to solve under typical rules. For 113, Wheeler suggests Γ(Γ(4))-(4!+4)/4. For 123, Wheeler suggests the expression:

<math> \sqrt{\sqrt{\sqrt{{(\sqrt{4}/.4)}^{4!}}}} - \sqrt{4} </math>

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