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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. Rouse Ball 6th edition published in 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 , This date aligns with the appearance of the problem in Knowledge: 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. Most also allow the factorial "! Typically the " log " operators or the successor function 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:. 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 "" 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 25, using typical rules.

Some alternate solutions are listed here, although there are actually many more correct solutions. The entries in blue are those that use four integers 4 rather than four digits 4 and the basic arithmetic operations.

Numbers without blue entries have no solution under these constraints. Additionally, solutions that repeat operators are marked in italics. Note that numbers with values less than one are not usually written with a leading zero. This is because "0" is a digit, and in this puzzle only the digit "4" can be used. A given number will generally have a few 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. Certain numbers, such as , are particularly difficult to solve under typical rules. This problem and its generalizations like the five fives and the six sixes problem, both shown below may be solved by a simple algorithm.

The basic ingredients are hash tables that map rationals to strings. In these tables, the keys are the numbers being represented by some admissible combination of operators and the chosen digit d , e.

There is one table for each number n of occurrences of d. Now there are two ways in which new entries may arise, either as a combination of existing ones through a binary operator, or by applying the factorial or square root operators which does not use additional instances of d. The first case is treated by iterating over all pairs of subexpressions that use a total of n instances of d. Memoization is used to ensure that every hash table is only computed once.

The second case factorials and roots is treated with the help of an auxiliary function, which is invoked every time a value v is recorded. This function computes nested factorials and roots of v up to some maximum depth, restricted to rationals.

The last phase of the algorithm consists in iterating over the keys of the table for the desired value of n and extracting and sorting those keys that are integers.

This algorithm was used to calculate the five fives and six sixes examples shown below. The more compact formula in the sense of number of characters in the corresponding value was chosen every time a key occurred more than once.

In the table below, the notation. From Wikipedia, the free encyclopedia. Mathematical Recreations and Essays, page 14 6th ed. Retrieved from " https: Views Read Edit View history. This page was last edited on 26 February , at By using this site, you agree to the Terms of Use and Privacy Policy.