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 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.
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
subfactorial, ("!" before the number: !4 equals 9),
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:
n = - ln[ln(sqrt(sqrt(...(sqrt(4))...))) / ln(4)] / ln(4)
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)
Note that numbers with values less than one are not usually written with a leading zero. For example, "0.4" is usually written as ".4". This is because "0" is a digit, and in this puzzle only the digit "4" can be used.
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:
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".
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