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. Mathematical games include many topics which are a part of recreational mathematics, but can also cover topics such as the mathematics of games, and playing games with mathematics. ... A number is an abstract entity that represents a count or measurement. ...

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. A mathematical expression is a string of symbols which describes (or expresses) a (potential or actual) computation using operators and operands. ... The whole numbers are the nonnegative integers (0, 1, 2, 3, ...) The set of all whole numbers is represented by the symbol = {0, 1, 2, 3, ...} Algebraically, the elements of form a commutative monoid under addition (with identity element zero), and under multiplication (with identity element one). ... This article discusses the number Four. ...

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 digit (".") and the square root operation, although sometimes square root is specifically excluded on the grounds that there is an implied "2" for the second root. 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 ("%"). Thus 4/4% = 100 and Γ(4)=6. A common use of the overline in this problem is for this value: 3 + 2 = 5 with apples, a popular choice in textbooks[1] Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. ... 5 - 2 = 3 Subtraction is one of the four basic arithmetic operations; it is essentially the opposite of addition. ... In mathematics, multiplication is an elementary arithmetic operation. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... For technical reasons, :) and some similar combinations starting with : redirect here. ... For factorial rings in mathematics, see unique factorisation domain. ... Exponentiation is a mathematical operation, written an, involving two numbers, the base a and the exponent n. ... In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ... The subfactorial function is used to calculate the number of permutations of a set of n objects in which none of the elements occur in their natural place, whereas the factorial function calculates the total number of permutations of the set. ... The Gamma function along part of the real axis In mathematics, the Gamma function is an extension of the factorial function to complex numbers. ...

Typically the "log" operators are not allowed, since there is 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: Above is the graph plots of Logarithms to various bases: is to base e, is to base 10, and is to base 1. ...

where the number of nested √ is 2n.

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. In a few cases some alternatives are shown, just to illustrate that there are alternatives, but many more alternatives are possible and not shown here:

• 0 = 44 − 44 = 4 − 4 + 4 − 4
• 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 = 4 + (4+4)/4
• 7 = 44/4 − 4 = 4 + 4 − (4/4)
• 8 = 4 + 4.4 − .4 = 4 + 4 + 4 - 4
• 9 = 4 + 4 + 4/4
• 10 = 44/4.4 = 4 + sqrt(4) + sqrt(4) + sqrt(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) = 4×4×4 / 4=4+4+4+4
• 17 = 4×4 + 4/4
• 18 = 44×.4 + .4 = 4×4 + 4 / sqrt(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.

A given number will generally have many possible solutions; any solution that meets the rules is acceptable. Some variations prefer the "fewest" number of operations, or prefer some operations to others. Others simply prefer "interesting" solutions, i.e., a surprising way to reach the goal.

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".

External links

• Bourke, Paul. Four Fours Problem.
• Wheeler, David A. 2002. The Definitive Four Fours Answer Key.
• John Valentine's Four Fours page.
• Cheng-Wen's Four Fours page.
• A. Bogomolny's Four Fours page at cut-the-knot
• Carver, Ruth. Four Fours Puzzle at MathForum.org
• Four Fours Problem at Wheels.org
• University of Toronto Math Net's The Four Fours Problem
• Marxen, Heiner. The Year Puzzle

• Riedel, Marko. A MAPLE program for the Four Fours problem (El problema de los cuatro cuatros, newsgroup es.ciencia.matematicas)
• Riedel, Marko. Solutions to the Five Fives and Six Sixes problem (El problema de los cinco cincos, newsgroup es.ciencia.matematicas)