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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