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Four Fours Puzzle - Solution
The Puzzle:
A popular mathematical pastime:
Use exactly four 4's to form every integer from 0 to 50, using only the operators +, −, ×, /, () (brackets), . (decimal point), √ (square root) and ! (factorial).
Example: 0 = 44 − 44
Our Solution:
Here is our list of solutions:
0 = 44 − 44
1 = 44 / 44 or (4 + 4)/(4 + 4) or (4/4) / (4/4) or ((4! − 4) / 4) − 4
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 × √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 = (√4 + 4!)/√4 + 4!
38 = 44 − 4!/4
39 = (4 × 4 − .4)/.4
40 = 44 − √(4 × 4)
41 = (√4 + 4!)/.4 − 4!
42 = √4 + 44 − 4
43 = 44 − 4/4
44 = 44.4 − .4
45 = 4/4 + 44
46 = 44 − √4 + 4
47 = 4! + 4! − 4/4
48 = 4 × (4 + 4 + 4)
49 = (4! − 4.4) / .4
50 = 4! / 4 + 44
BONUS! 51 to 100, but we have gaps. Can you fill them in?
51 = (4! − 4 + .4) / .4
52 = 44 + 4 + 4
53
54 = 4! + 4! + (4! / 4)
55
56 = 44 + 4! / √4
57
58 = 4! + 4! + 4/.4
59 = 4!/.4 − 4/4
60 = 44 + 4 × 4
61 = 4!/.4 + 4/4
62 = (4 + 4) ^ √4 − √4
63 = (4! + √4) / .4 − √4
64 = 4 × 4 × √4 × √4
65 = (4^4 +4) / 4
66 = (4 + 4) ^ (√4) + √4
67 = (4! + √4) / .4 + √4
68 = (4 + 4) ^ √4 + 4
69 = (4! + 4 − .4) / .4
70 = 44 + 4! + √4
71 = (4 + 4! + .4) / .4
72 = (4! / 4) ^ √4 × √4
73
74 = 4! + 4! + 4! + √4
75 = (4! + √4 + 4) / .4
76 = 4! × 4 − 4! + 4
77
78
79
80 = 4! × 4 − 4×4
81
82 = 4! / .4 + 4! − √4
83
84 = 4! × 4 − 4! / √4
85
86 = 4! × 4 − 4/.4
87
88 = 4! × 4 − 4 − 4
89 = 4! + (4! + √4) / .4
90 = 4! × 4 − 4! / 4
91 = 4 × 4! − √4 / .4
92 = 4! × (√4 + √4) − 4
93
94 = 4! × 4 − 4/√4
95 = 4! × 4 − 4/4
96 = 4! × 4 + 4 − 4
97 = 4! × 4 + 4/4
98 = 4! × 4 + 4/√4
99
100 = 4! × 4 + √4 + √4
(Note: people have managed to do this for 1000s of numbers)
0 = 44 − 44
1 = 44 / 44 or (4 + 4)/(4 + 4) or (4/4) / (4/4) or ((4! − 4) / 4) − 4
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 × √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 = (√4 + 4!)/√4 + 4!
38 = 44 − 4!/4
39 = (4 × 4 − .4)/.4
40 = 44 − √(4 × 4)
41 = (√4 + 4!)/.4 − 4!
42 = √4 + 44 − 4
43 = 44 − 4/4
44 = 44.4 − .4
45 = 4/4 + 44
46 = 44 − √4 + 4
47 = 4! + 4! − 4/4
48 = 4 × (4 + 4 + 4)
49 = (4! − 4.4) / .4
50 = 4! / 4 + 44
BONUS! 51 to 100, but we have gaps. Can you fill them in?
51 = (4! − 4 + .4) / .4
52 = 44 + 4 + 4
53
54 = 4! + 4! + (4! / 4)
55
56 = 44 + 4! / √4
57
58 = 4! + 4! + 4/.4
59 = 4!/.4 − 4/4
60 = 44 + 4 × 4
61 = 4!/.4 + 4/4
62 = (4 + 4) ^ √4 − √4
63 = (4! + √4) / .4 − √4
64 = 4 × 4 × √4 × √4
65 = (4^4 +4) / 4
66 = (4 + 4) ^ (√4) + √4
67 = (4! + √4) / .4 + √4
68 = (4 + 4) ^ √4 + 4
69 = (4! + 4 − .4) / .4
70 = 44 + 4! + √4
71 = (4 + 4! + .4) / .4
72 = (4! / 4) ^ √4 × √4
73
74 = 4! + 4! + 4! + √4
75 = (4! + √4 + 4) / .4
76 = 4! × 4 − 4! + 4
77
78
79
80 = 4! × 4 − 4×4
81
82 = 4! / .4 + 4! − √4
83
84 = 4! × 4 − 4! / √4
85
86 = 4! × 4 − 4/.4
87
88 = 4! × 4 − 4 − 4
89 = 4! + (4! + √4) / .4
90 = 4! × 4 − 4! / 4
91 = 4 × 4! − √4 / .4
92 = 4! × (√4 + √4) − 4
93
94 = 4! × 4 − 4/√4
95 = 4! × 4 − 4/4
96 = 4! × 4 + 4 − 4
97 = 4! × 4 + 4/4
98 = 4! × 4 + 4/√4
99
100 = 4! × 4 + √4 + √4
(Note: people have managed to do this for 1000s of numbers)