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

You are watching: Four fours answers 1-100

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) = 1Cannot 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, divisionsqrt,^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.

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


0
*
" hspace="45" width="77" height="15" align="absmiddle">
59
*
" hspace="45" width="59" height="30" align="absmiddle">
1
*
" hspace="45" width="29" height="30" align="absmiddle">
60
*
" hspace="45" width="62" height="15" align="absmiddle">
2
*
" hspace="45" width="51" height="30" align="absmiddle">
61
*
" hspace="45" width="59" height="30" align="absmiddle">
3
*
" hspace="45" width="78" height="30" align="absmiddle">
62
*
" hspace="45" width="104" height="32" align="absmiddle">
4
*
" hspace="45" width="98" height="15" align="absmiddle">
63
*
" hspace="45" width="104" height="35" align="absmiddle">
5
*
" hspace="45" width="78" height="30" align="absmiddle">
64
*
" hspace="45" width="37" height="23" align="absmiddle">
6
*
" hspace="45" width="78" height="30" align="absmiddle">
65
*
" hspace="45" width="75" height="35" align="absmiddle">
7
*
" hspace="45" width="64" height="30" align="absmiddle">
66
*
" hspace="45" width="68" height="30" align="absmiddle">
8
*
" hspace="45" width="76" height="30" align="absmiddle">
67
*
" hspace="45" width="104" height="35" align="absmiddle">
9
*
" hspace="45" width="64" height="30" align="absmiddle">
68
*
" hspace="45" width="49" height="33" align="absmiddle">
10
*
" hspace="45" width="50" height="30" align="absmiddle">
69
*
" hspace="45" width="88" height="35" align="absmiddle">
11
*
" hspace="45" width="59" height="30" align="absmiddle">
70
*
" hspace="45" width="122" height="19" align="absmiddle">
12
*
" hspace="45" width="99" height="15" align="absmiddle">
71
*
" hspace="45" width="67" height="30" align="absmiddle">
13
*
" hspace="45" width="78" height="35" align="absmiddle">
72
*
" hspace="45" width="58" height="35" align="absmiddle">
14
*
" hspace="45" width="104" height="30" align="absmiddle">
73
*
" hspace="45" width="127" height="50" align="absmiddle">
15
*
" hspace="45" width="62" height="30" align="absmiddle">
74
*
" hspace="45" width="122" height="19" align="absmiddle">
16
*
" hspace="45" width="77" height="15" align="absmiddle">
75
*
" hspace="45" width="88" height="35" align="absmiddle">
17
*
" hspace="45" width="79" height="30" align="absmiddle">
76
*
" hspace="45" width="85" height="30" align="absmiddle">
18
*
" hspace="45" width="92" height="15" align="absmiddle">
77
*
" hspace="45" width="86" height="41" align="absmiddle">
19
*
" hspace="45" width="72" height="30" align="absmiddle">
78
*
" hspace="45" width="104" height="30" align="absmiddle">
20
*
" hspace="45" width="67" height="30" align="absmiddle">
79
*
" hspace="45" width="102" height="41" align="absmiddle">
21
*
" hspace="45" width="72" height="30" align="absmiddle">
80
*
" hspace="45" width="84" height="30" align="absmiddle">
22
*
" hspace="45" width="88" height="35" align="absmiddle">
81
*
" hspace="45" width="63" height="31" align="absmiddle">
23
*
" hspace="45" width="88" height="30" align="absmiddle">
82
*
" hspace="45" width="97" height="30" align="absmiddle">
24
*
" hspace="45" width="76" height="35" align="absmiddle">
83
*
" hspace="45" width="102" height="41" align="absmiddle">
25
*
" hspace="45" width="88" height="30" align="absmiddle">
84
*
" hspace="45" width="69" height="19" align="absmiddle">
26
*
" hspace="45" width="84" height="30" align="absmiddle">
85
*
" hspace="45" width="63" height="38" align="absmiddle">
27
*
" hspace="45" width="126" height="60" align="absmiddle">
86
*
" hspace="45" width="85" height="19" align="absmiddle">
28
*
" hspace="45" width="80" height="15" align="absmiddle">
87
*
" hspace="45" width="79" height="32" align="absmiddle">
29
*
" hspace="45" width="72" height="30" align="absmiddle">
88
*
" hspace="45" width="50" height="15" align="absmiddle">
30
*
" hspace="45" width="101" height="19" align="absmiddle">
89
*
" hspace="42" width="91" height="45" align="absmiddle">
31
*
" hspace="45" width="104" height="35" align="absmiddle">
90
*
" hspace="45" width="85" height="19" align="absmiddle">
32
*
" hspace="45" width="42" height="33" align="absmiddle">
91
*
" hspace="45" width="87" height="35" align="absmiddle">
33
*
" hspace="45" width="88" height="35" align="absmiddle">
92
*
" hspace="45" width="69" height="19" align="absmiddle">
34
*
" hspace="45" width="101" height="19" align="absmiddle">
93
*
" hspace="45" width="104" height="46" align="absmiddle">
35
*
" hspace="45" width="97" height="40" align="absmiddle">
94
*
" hspace="45" width="127" height="39" align="absmiddle">
36
*
" hspace="45" width="85" height="15" align="absmiddle">
95
*
" hspace="45" width="83" height="35" align="absmiddle">
37
*
" hspace="45" width="97" height="40" align="absmiddle">
96
38
*
" hspace="45" width="114" height="19" align="absmiddle">
97
39
*
" hspace="45" width="66" height="35" align="absmiddle">
98
40
*
" hspace="45" width="128" height="21" align="absmiddle">
99
41
*
" hspace="45" width="97" height="35" align="absmiddle">
100
42
*
" hspace="45" width="66" height="35" align="absmiddle">
101
43
*
" hspace="45" width="50" height="30" align="absmiddle">
102
44
*
" hspace="45" width="49" height="30" align="absmiddle">
103
45
*
" hspace="45" width="50" height="30" align="absmiddle">
104
46
*
" hspace="45" width="110" height="19" align="absmiddle">
105
47
*
" hspace="45" width="87" height="30" align="absmiddle">
106
48
*
" hspace="45" width="84" height="30" align="absmiddle">
107
49
*
" hspace="45" width="95" height="30" align="absmiddle">
108
50
*
" hspace="45" width="57" height="30" align="absmiddle">
109
51
*
" hspace="45" width="88" height="35" align="absmiddle">
110
52
*
" hspace="45" width="63" height="15" align="absmiddle">
111
53
*
" hspace="45" width="97" height="35" align="absmiddle">
112
54
*
" hspace="45" width="110" height="19" align="absmiddle">
113
55
*
" hspace="45" width="75" height="35" align="absmiddle">
114
56
*
" hspace="45" width="94" height="15" align="absmiddle">
115
57
*
" hspace="45" width="104" height="35" align="absmiddle">
116
58
*
" hspace="45" width="111" height="35" align="absmiddle">

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.

See more: What Is One Third In Percentage S, 1/3 As A Percent

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.