Here are my solutions for the first few numbers. Some of these are not optimal (fewest operators):
1. 44/44
2. 4/4+4/4
3. (4+4+4)/4
4. 4*(4-4)+4
5. 4!/4-4/4 (not optimal)
6. (4+4)/4+4
7. 44/4-4
8. 4+4+4-4
9. 4+4+4/4
10 . 4+4+4-sqrt(4) (not optimal)
11. 4!/sqrt(4)-4/4 (not optimal)
12. 4!-4-4-4 (not optimal)
13. 4!/sqrt(4)+4/4 (not optimal)
14. 4+4+4+sqrt(4)
15. 4*4-4/4 (not optimal)
16. 4+4+4+4
17. 4*4+4/4
18. 4*4+4/sqrt(4) (better than the solution at Math Forum - Four 4's Puzzle)
19. 4!-4-4/4
20. 4!+4-4-4
Wednesday, July 07, 2004
123456789 = 100
Puzzle: I received email asking me to make the numbers 123456789 add up to 100, by inserting any of these arithmetic operations (+ - * /) into the string of numbers (and keeping the numbers in the above order). I've seen this in a book somewhere, and I think the book gave only one solution. You might want to try it.
Solution: I found this solution by hand: 1+2+3-4+5+6+78+9, and my computer (with the standard computer rules of precendence, like 1/2*34 is 17) found all of these solutions:
1 123+45-67+8-9
2 123+4-5+67-89
3 123+4*5-6*7+8-9
4 123-45-67+89
5 123-4-5-6-7+8-9
6 12+34+5*6+7+8+9
7 12+34-5+6*7+8+9
8 12+34-5-6+7*8+9
9 12+34-5-6-7+8*9
10 12+3+4+5-6-7+89
11 12+3+4-56/7+89
12 12+3-4+5+67+8+9
13 12+3*45+6*7-89
14 12+3*4+5+6+7*8+9
15 12+3*4+5+6-7+8*9
16 12+3*4-5-6+78+9
17 12-3+4*5+6+7*8+9
18 12-3+4*5+6-7+8*9
19 12-3-4+5-6+7+89
20 12-3-4+5*6+7*8+9
21 12-3-4+5*6-7+8*9
22 12*3-4+5-6+78-9
23 12*3-4-5-6+7+8*9
24 12*3-4*5+67+8+9
25 12/3+4*5-6-7+89
26 12/3+4*5*6-7-8-9
27 12/3+4*5*6*7/8-9
28 12/3/4+5*6+78-9
29 1+234-56-7-8*9
30 1+234*5*6/78+9
31 1+234*5/6-7-89
32 1+23-4+56+7+8+9
33 1+23-4+56/7+8*9
34 1+23-4+5+6+78-9
35 1+23-4-5+6+7+8*9
36 1+23*4+56/7+8-9
37 1+23*4+5-6+7-8+9
38 1+23*4-5+6+7+8-9
39 1+2+34-5+67-8+9
40 1+2+34*5+6-7-8*9
41 1+2+3+4+5+6+7+8*9
42 1+2+3-45+67+8*9
43 1+2+3-4+5+6+78+9
44 1+2+3-4*5+6*7+8*9
45 1+2+3*4-5-6+7+89
46 1+2+3*4*56/7-8+9
47 1+2+3*4*5/6+78+9
48 1+2-3*4+5*6+7+8*9
49 1+2-3*4-5+6*7+8*9
50 1+2*34-56+78+9
51 1+2*3+4+5+67+8+9
52 1+2*3+4*5-6+7+8*9
53 1+2*3-4+56/7+89
54 1+2*3-4-5+6+7+89
55 1+2*3*4*5/6+7+8*9
56 1-23+4*5+6+7+89
57 1-23-4+5*6+7+89
58 1-23-4-5+6*7+89
59 1-2+3+45+6+7*8-9
60 1-2+3*4+5+67+8+9
61 1-2+3*4*5+6*7+8-9
62 1-2+3*4*5-6+7*8-9
63 1-2-34+56+7+8*9
64 1-2-3+45+6*7+8+9
65 1-2-3+45-6+7*8+9
66 1-2-3+45-6-7+8*9
67 1-2-3+4*56/7+8*9
68 1-2-3+4*5+67+8+9
69 1-2*3+4*5+6+7+8*9
70 1-2*3-4+5*6+7+8*9
71 1-2*3-4-5+6*7+8*9
72 1*234+5-67-8*9
73 1*23+4+56/7*8+9
74 1*23+4+5+67-8+9
75 1*23-4+5-6-7+89
76 1*23-4-56/7+89
77 1*23*4-56/7/8+9
78 1*2+34+56+7-8+9
79 1*2+34+5+6*7+8+9
80 1*2+34+5-6+7*8+9
81 1*2+34+5-6-7+8*9
82 1*2+34-56/7+8*9
83 1*2+3+45+67-8-9
84 1*2+3+4*5+6+78-9
85 1*2+3-4+5*6+78-9
86 1*2+3*4+5-6+78+9
87 1*2-3+4+56/7+89
88 1*2-3+4-5+6+7+89
89 1*2-3+4*5-6+78+9
90 1*2*34+56-7-8-9
91 1*2*3+4+5+6+7+8*9
92 1*2*3-45+67+8*9
93 1*2*3-4+5+6+78+9
94 1*2*3-4*5+6*7+8*9
95 1*2*3*4+5+6+7*8+9
96 1*2*3*4+5+6-7+8*9
97 1*2*3*4-5-6+78+9
98 1*2/3+4*5/6+7+89
99 1/2*34-5+6-7+89
100 1/2*3/4*56+7+8*9
101 1/2/3*456+7+8+9
The shortest solution (fewest operations) is 123-45-67+89. There are more solutions, if a negative sign (which is different from subtraction) is allowed. Allowing parentheses would produce a lot more solutions.
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