There is an old math problem in which you are challenged to use four 4s and any operations to write equations that have the numbers from 0 to 100 as the answer.
Shekhar Dutta has shared his solutions below. He uses addition, subtraction, multiplication, division, power, root, factorial, double factorial,
logic gates, bitwise shift, gamma function, brackets, decimal, continuing decimal, continuation, reciprocal and all other mathematical functions and constants.
Decimals, that is, 4.4 can be used. Continuing decimals, that is, .4~ can be used. It means 0.4444444... or 4/9. Continuation, that
is, 44, 444, 4444, etc. can be used. Reciprocals, that is, rec(.4~) can be used. This means rec(4/9) = 9/4. This means dividing 1 by the number
that is in the bracket. Greatest integer and lowest integer can also be used. Grtint (4.4) means the greatest integer below 4.4, that is, 4.
Lowint (4.4) means the lowest integer above 4.4, that is 5. Also, Shekhar allows Logarithms and percentage (4% means 4/100).
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)
21 = (4.4+4)/.4
22 = 44*sqrt(4)/4
23 = (4*4!-4)/4
24 = 4*4+4+4
25 = (4*4!+4)/4
26 = 4/.4+4*4
27 = 4-4/4+4!
28 = 44-4*4
29 = 4/.4/.4+4
30 = (4+4+4)/.4
31 = (4!+ 4)/4+4!
32 = 4*4+4*4
33 = (4-.4)/.4+4!
34 = 44-4/.4
35 = 44/4+4!
36 = 44-4-4
37 = (sqrt(4)+4!)/sqrt(4)+4!
38 = 44-4!/4
39 = (4*4-.4)/.4
40 = 44-sqrt(4*4)
41 = (sqrt(4)+4!)/.4-4!
42 = sqrt(4)+44-4
43 = 44-4/4
44 = 44.4-.4
45 = 4/4+44
46 = 44-sqrt(4)+4
47 = 4!+4!-4/4
48 = 4*(4+4+4)
49 = (4!-4.4)/.4
50 = 4!/4+44
51 = (4!-sqrt(4))/.4-4
52 = 4+4+44
53 = sqrt(4)/.4+4!+4!
54 = 4/.4+44
55 = 44/(.4+.4)
56 = 4*(4/.4+4)
57 = (4!-.4)/.4-sqrt(4)
58 = (4^4-4!)/4
59 = 4!/.4-4/4
60 = 4*4+44
61 = 4!/.4+4/4
62 = (.4+.4+4!)/.4
63 = (4^4-4)/4
64 = 4!-4+44
65 = (4^4+4)/4
66 = (4!+4)/.4-4
67 = (sqrt(4)+4!)/.4+sqrt(4)
68 = 4*4*4+4
69 = (4-.4+4!)/.4
70 = 4!/.4+4/.4
71 = (4!+4.4)/.4
72 = 4!+44+4
73 = sqrt(sqrt(sqrt(4)^4!))+4/.4~
74 = (4!+4)/.4+4
75 = 4!/(.4+.4)/.4
76 = 4!/.4+4*4
77 = sqrt(4/.4~)^4-4
78 = 4*(4!-4)-sqrt(4)
79 = (4!-sqrt(4))/.4+4!
80 = 4*(4*4+4)
81 = (4/4-4)^4
82 = 4*(4!-4)+sqrt(4)
83 = (4!-.4)/.4+4!
84 = 44*sqrt(4)-4
85 = (4/.4+4!)/.4
86 = 44/.4-4!
87 = 4*4!-4/.4~
88 = 44+44
89 = (sqrt(4)+4!)/.4+4!
90 = 44*sqrt(4)+sqrt(4)
91 = 4*4!-sqrt(4)/.4
92 = 44*sqrt(4)+4
93 = 4*4!-sqrt(4/.4~)
94 = 4*(4!-.4)-.4
95 = 4*4!-4/4
96 = 4*(4.4-.4)!
97 = 4*4!+4/4
98 = 4*(44/44 !+.4)+.4
99 = 44/sqrt(.4~*.4~)
100 = 44/.44
Many thanks to Chang Yao Liu for correcting the answers to #8, #9, and #72.
Click here for the definitive solution key from 0 to 40000 by David Wheeler.
If you like this puzzle, then try the
New Year Challenge .