GeekDad Puzzle of the Week Solution: Palindromic Sums of Squares

(NOTE: For purposes of this puzzle, please consider only sums of sets of squares of positive integers.) Hopefully, a lot of people remembered that the sum of numbers from 1^2 + 2^2 + ... n^2 = n(n+1)(2n+1)/6. Otherwise, a lots of processors may have done a lot of unnecessary math!
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8 + 18 + 9 + 4 + 1 = 40 squares!

This past week's puzzle, as posted:

If you have been on Facebook at all over the last year, you have probably seen the "puzzle" where there is a grid, and the question reads "How many squares do you see?" Some people (clearly not readers of this blog) only count the 1×1 squares, and not the 2×2, 3×3, 4×4 or larger squares.

Similarly, some numbers, when squared, are palindromes: 112 is 121, and 262 is 676. Other numbers are the sums of sets of consecutive squares: 92 + 102 = 181, and 42 + 52 + 62 = 77.

This week’s puzzle is simple: How many numbers between 1 million and 10 million are both palindromes and the sums of sets of consecutive squares?

(NOTE: For purposes of this puzzle, please consider only sums of sets of squares of positive integers.)

Hopefully, a lot of people remembered that the sum of numbers from 12 + 22 + ... n2 = n(n+1)(2n+1)/6. Otherwise, a lots of processors may have done a lot of unnecessary math!

There are some 51 different numbers between 1 million and 10 millions that are both palindromes and the sums of consecutive squares. Note that one of these numbers (9343439) can be expressed as the sum of two different sets of sums, and there are some five numbers that when squared by themselves are 7-digit palindromes.

The largest set of numbers was 9313139 = 262 + 272 + ... + 3032, with 278 different terms! The smallest was a five-way tie, with 1690961, 3162613, 3187813, 5258525, and 5824285 all expressed as the sum of two consecutive squares.

Here are all the numbers, with their component starting and ending squares:

  1. 1077701 = 632 + ... + 1512 (89 terms)
  2. 1224221 = 1732 + ... + 2062 (34 terms)
  3. 1365631 = 342 + ... + 1602 (127 terms)
  4. 1681861 = 1562 + ... + 2062 (51 terms)
  5. 1690961 = 9192 + 9202 (2 terms)
  6. 1949491 = 1062 + ... + 1912 (86 terms)
  7. 1972791 = 1642 + ... + 2172 (54 terms)
  8. 1992991 = 12 + ... + 1812 (181 terms)
  9. 2176712 = 1892 + ... + 2362 (48 terms)
  10. 2904092 = 5992 + ... + 6062 (8 terms)
  11. 3015103 = 272 + ... + 2082 (182 terms)
  12. 3162613 = 12572 + 12582 (2 terms)
  13. 3187813 = 12622 + 12632 (2 terms)
  14. 3242423 = 172 + ... + 2132 (197 terms)
  15. 3628263 = 1022 + ... + 2282 (127 terms)
  16. 4211124 = 1722 + ... + 2602 (89 terms)
  17. 4338334 = 6232 + ... + 6332 (11 terms)
  18. 4424244 = 1282 + ... + 2482 (121 terms)
  19. 4776774 = 1012 + ... + 2482 (148 terms)
  20. 5090905 = 7092 + ... + 7182 (10 terms)
  21. 5258525 = 16212 + 16222 (2 terms)
  22. 5276725 = 2102 + ... + 2922 (83 terms)
  23. 5367635 = 732 + ... + 2542 (182 terms)
  24. 5479745 = 362 + ... + 2542 (219 terms)
  25. 5536355 = 4152 + ... + 4442 (30 terms)
  26. 5588855 = 2262 + ... + 3042 (79 terms)
  27. 5603065 = 2612 + ... + 3252 (65 terms)
  28. 5718175 = 1462 + ... + 2722 (127 terms)
  29. 5824285 = 17062 + 17072 (2 terms)
  30. 6106016 = 742 + ... + 2652 (192 terms)
  31. 6277726 = 462 + ... + 2662 (221 terms)
  32. 6523256 = 6312 + ... + 6462 (16 terms)
  33. 6546456 = 542 + ... + 2702 (217 terms)
  34. 6780876 = 8642 + ... + 8722 (9 terms)
  35. 6831386 = 7492 + ... + 7602 (12 terms)
  36. 6843486 = 542 + ... + 2742 (221 terms)
  37. 6844486 = 1592 + ... + 2902 (132 terms)
  38. 7355537 = 2242 + ... + 3212 (98 terms)
  39. 8424248 = 222 + ... + 2932 (272 terms)
  40. 9051509 = 17362 + 17372 + 17382 (3 terms)
  41. 9072709 = 6822 + ... + 7002 (19 terms)
  42. 9105019 = 722 + ... + 3022 (231 terms)
  43. 9313139 = 262 + ... + 3032 (278 terms)
  44. 9343439 = 6572 + ... + 6772 (21 terms)
  45. 9334339 = 4772 + ... + 5142 (38 terms)
  46. 9435349 = 1672 + ... + 3202 (154 terms)
  47. 9563659 = 8202 + ... + 8332 (14 terms)
  48. 9793979 = 4822 + ... + 5202 (39 terms)
  49. 9814189 = 1722 + ... + 3252 (154 terms)
  50. 9838389 = 2232 + ... + 3432 (121 terms)
  51. 9940499 = 1362 + ... + 3182 (183 terms)

The number 9343439 can alternately be expressed as 1022 + ... + 3072 (206 terms.)

Congratulations to Gregory Hyung Jin Park for submitting this week's randomly chosen answer, and winning this week's $50 ThinkGeek Gift Certificate. For everyone else, please feel free to use coupon code GEEKDAD23CD for $10 off a ThinkGeek order of $50 or more.