### Four Fours

 Every now and then I get addicted by a puzzle. Here's the first that I was ever addicted to. It was given to me my one of my maths teachers when I was a teenager.

The goal is to just use four four's to represent all the integer numbers from 1 to 64. you can use whatever mathematical symbols you want to derive your answers. In the notation below, sqrt(x) is the square root of x and pow(x, y) is x to the power of y.

Over the years (and without the aid of a computer), I reduced the number of missing entries to 9.

Here's my solution. Digitized at last.

1. (4 + 4) / (4 + 4)
2. (4 / 4) + (4 / 4)
3. 4 - sqrt(4) + (4 / 4)
4. (4 \* 4) / (sqrt(4) + sqrt(4))
5. 4 + sqrt(4) - (4 / 4)
6. 4 + sqrt(4) + sqrt(4) - sqrt(4)
7. 4 + sqrt(4) + (4 / 4)
8. 4 + 4 + 4 - 4
9. 4 + 4 + (4 / 4)
10. 4 + sqrt(4) + sqrt(4) + sqrt(4)
11. (4! / sqrt(4)) - (4 / 4)
12. (4 \* 4) - sqrt(4) - sqrt(4)
13. (4! / sqrt(4)) + (4 / 4)
14. (4! / (4 - sqrt(4)) + sqrt(4)
15. (4 \* 4) - (4 / 4)
16. sqrt(4) \* sqrt(4) \* sqrt(4) \* sqrt(4)
17. (4 \* 4) + (4 / 4)
18. (4 \* 4) + 4 - sqrt(4)
19. 4! - 4 - (4 / 4)
20. 4 \* sqrt(4! + (4 / 4))
21. 4! + (4 / 4) - 4
22. 4! - 4 + 4 - sqrt(4)
23. 4! + (4 / 4) - sqrt(4)
24. (4! \* (4 / sqrt(4))) / sqrt(4)
25. 4! + sqrt(4) - (4 / 4)
26. 4! + 4 - sqrt(4)
27. 4! + 4 - (4 / 4)
28. 4! + sqrt(4) + (4 / sqrt(4))
29. 4! + 4 + (4 / 4)
30. 4! + sqrt(4) + sqrt(4) + sqrt(4)
31. (pow(sqrt(sqrt(sqrt(4))), 4!) - sqrt(4)) / sqrt(4)
32. (4 \* 4) + (4 \* 4)
33. (pow(sqrt(sqrt(sqrt(4))), 4!) + sqrt(4)) / sqrt(4)
34. 4! + 4 + 4 + sqrt(4)
35. 4! + (4! - sqrt(4)) / sqrt(4)
36. 4! + 4 + 4 + 4
37. 4! + (4! + sqrt(4)) / sqrt(4)
38. 4! + (4 \* 4) - sqrt(4)
39. no solution
40. (4! \* sqrt(4)) - 4 - 4
41. sqrt(((4 + 4)! + 4!) / 4!)
42. (4! \* sqrt(4)) - 4 - sqrt(4)
43. no solution
44. (4! \* sqrt(4)) - sqrt(4) - sqrt(4)
45. no solution
46. (4! \* sqrt(4)) - 4 + sqrt(4)
47. (4! \* sqrt(4)) - (4 / 4)
48. (4! \* sqrt(4)) + 4 - 4
49. (4! \* sqrt(4)) + (4 / 4)
50. (4! \* sqrt(4)) + 4 - sqrt(4)
51. no solution
52. (4! \* sqrt(4)) + sqrt(4) + sqrt(4)
53. no solution
54. (4! \* sqrt(4)) + sqrt(4) + 4
55. no solution
56. (4! \* sqrt(4)) + 4 + 4
57. no solution
58. pow(sqrt(sqrt(sqrt(4))), 4!) - 4 - sqrt(4)
59. no solution
60. pow(sqrt(sqrt(sqrt(4))), 4!) - sqrt(4) - sqrt(4)
61. no solution
62. pow(sqrt(sqrt(sqrt(4))), 4!) - (4 / sqrt(4))
63. pow(sqrt(sqrt(sqrt(4))), 4!) - (4/4)
64. 4\*4\*sqrt(4)\*sqrt(4)

I see that Carl Johansen was addicted to this too, and even wrote a C++ program to solve it. He's gone to 100 and also used .4bar (.44444...), which wasn't something I incorporated.

It's nice to see the puzzle finally solved though.

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