The Four Fours Problem

Solutions from 0 to 116, not including 113 which is probably not findable by the rules given.

This puzzle is most commonly formulated as follows:

"Given no more than 4 instances of the digit "4", represent all integers using a finite number of mathematical symbols and operators in common use."

Notes on the Rules:

• You cannot use representations for other numbers, eg: use of π in sin(π / 2) = 1
• Cannot use ... to represent an infinite number of operations eg: sqrt(sqrt(sqrt(...4))) = 1
• Obviously negative integers can be represented if a positive number can.
• It can be proved that not all integers can be represented.

Subset of acceptable symbols and operators:
+,- addition subtraction
*,/ multiplication, division
sqrt,^ square root, power
! factorial, eg: n! = n * (n-1) * (n-2) * ... * 2 * 1
. decimal point, eg: .4
′ repeating decimal, eg: .4′ = .444444444...

With other rules, other answers are possible. Some variations allow you to use any number of 4s up to 4. Others allow more obscure mathematical operators (gamma functions, for example). It can be easily demonstrated that, even by the conservative ruleset that I describe above, some numbers can be generated an infinite number of different ways. The presence of unary operators means that there are an infinite number of different patterns to search for the answers to these problems, which makes computer solutions an interesting problem, which will never have a perfect solution.

I found the following answers the old-fashioned way, by hand search and logic. Though some people put a premium on simple formulations, I have in several places listed answers that were for some reason amusing to me, or surprisingly complex. My answer to 73 is a good example of amusing complexity, as is 27. Since all the answers are integers, I started the search by generating a list of integers that could be build with one or two 4s. These became useful building blocks in finding the larger numbers. Since it is possible to create an infinite number of integers from even a single four (for example, 4! = 24. 4!! = 620448401733239439360000, and so forth). I listed only the smaller integers in my reference table, knowing that these were most likely to be useful.

For your convenience, I present a table of useful 1- and 2-4 combinations.

0		59
1		60
2		61
3		62
4		63
5		64
6		65
7		66
8		67
9		68
10		69
11		70
12		71
13		72
14		73
15		74
16		75
17		76
18		77
19		78
20		79
21		80
22		81
23		82
24		83
25		84
26		85
27		86
28		87
29		88
30		89
31		90
32		91
33		92
34		93
35		94
36		95
37		96
38		97
39		98
40		99
41		100
42		101
43		102
44		103
45		104
46		105
47		106
48		107
49		108
50		109
51		110
52		111
53		112
54		113
55		114
56		115
57		116
58

A note to math teachers: I have gotten abusive e-mails from math teachers who feel that this page is essentially a cheat sheet, giving away the answers to a puzzle they would like to give as homework. Frankly, this is a widely known puzzle that I first heard in the Reagan administration and it wasn't new then. Not only are solutions widely known, but software that allows the automated generation of solutions has also been released into the public domain (not that I have used any such software above). What I'm trying to say is it’s time to find new assignments. Assigning the four fours problem in a math class is at this point like asking student to predict the end of the vietnam war in current events class — it might have been a great question in 1970 but it doesn’t test what you want it to anymore.

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