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"Partridge numbers" with partial completion
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| references: | |
| https://www.facebook.com/groups/391950357895182/permalink/1748946985528839/ | |
| https://www.puzzlefun.online/problems | |
| https://erich-friedman.github.io/mathmagic/0802.html | |
| For each N, try to pack 1 1x1 square, 2 2x2 squares, 3 3x3 squares, 4 4x4 squares, ..., N NxN squares into a N*(N+1)/2xN*(N+1)/2 square | |
| The goal is to omit as little area as possible. (e.g. omitting 1 2x2 and 2 3x3s means that a total area of 4+9+9 is omitted) | |
| Here are the optimal results for small N: | |
| N=1 -> 0 area omitted | |
| 1 | |
| N=2 -> 4 area omitted | |
| 100 | |
| 022 | |
| 022 | |
| N=3 -> 4 area omitted | |
| 100333 | |
| 022333 | |
| 022333 | |
| 333333 | |
| 333333 | |
| 333333 | |
| N=4 -> 16 area omitted | |
| 1000000333 | |
| 0000000333 | |
| 0004444333 | |
| 3334444333 | |
| 3334444333 | |
| 3334444333 | |
| 2244444444 | |
| 2244444444 | |
| 2244444444 | |
| 2244444444 | |
| N=5 -> 13 area omitted | |
| 100000000004444 | |
| 555555555504444 | |
| 555555555504444 | |
| 555555555504444 | |
| 555555555522333 | |
| 555555555522333 | |
| 444444444444333 | |
| 444444444444333 | |
| 444444444444333 | |
| 444444444444333 | |
| 555555555555555 | |
| 555555555555555 | |
| 555555555555555 | |
| 555555555555555 | |
| 555555555555555 | |
| N=6 -> 8 area omitted | |
| 100055555666666333333 | |
| 444455555666666333333 | |
| 444455555666666333333 | |
| 444455555666666666666 | |
| 444455555666666666666 | |
| 555554444666666666666 | |
| 555554444666666666666 | |
| 555554444666666666666 | |
| 555554444666666666666 | |
| 555550333666666666666 | |
| 666666333666666666666 | |
| 666666333666666666666 | |
| 666666044444444666666 | |
| 666666044444444666666 | |
| 666666044444444666666 | |
| 666666044444444666666 | |
| 555555555555555666666 | |
| 555555555555555666666 | |
| 555555555555555666666 | |
| 555555555555555666666 | |
| 555555555555555666666 | |
| N=7 -> 8 area omitted (solution is same as N=6 but with 7s added) | |
| 1000555556666663333337777777 | |
| 4444555556666663333337777777 | |
| 4444555556666663333337777777 | |
| 4444555556666666666667777777 | |
| 4444555556666666666667777777 | |
| 5555544446666666666667777777 | |
| 5555544446666666666667777777 | |
| 5555544446666666666667777777 | |
| 5555544446666666666667777777 | |
| 5555503336666666666667777777 | |
| 6666663336666666666667777777 | |
| 6666663336666666666667777777 | |
| 6666660444444446666667777777 | |
| 6666660444444446666667777777 | |
| 6666660444444446666667777777 | |
| 6666660444444446666667777777 | |
| 5555555555555556666667777777 | |
| 5555555555555556666667777777 | |
| 5555555555555556666667777777 | |
| 5555555555555556666667777777 | |
| 5555555555555556666667777777 | |
| 7777777777777777777777777777 | |
| 7777777777777777777777777777 | |
| 7777777777777777777777777777 | |
| 7777777777777777777777777777 | |
| 7777777777777777777777777777 | |
| 7777777777777777777777777777 | |
| 7777777777777777777777777777 | |
| 9 | |
| N=8 -> 0 area omitted | |
| 444422555556666666666666666667777777 | |
| 444422555556666666666666666667777777 | |
| 444422555556666666666666666667777777 | |
| 444422555556666666666666666667777777 | |
| 666666555556666666666666666667777777 | |
| 666666444416666666666666666667777777 | |
| 666666444433388888888888888887777777 | |
| 666666444433388888888888888887777777 | |
| 666666444433388888888888888887777777 | |
| 666666777777788888888888888887777777 | |
| 666666777777788888888888888887777777 | |
| 666666777777788888888888888887777777 | |
| 666666777777788888888888888887777777 | |
| 666666777777788888888888888887777777 | |
| 666666777777788888888888888887777777 | |
| 666666777777788888888888888887777777 | |
| 888888885555588888888888888887777777 | |
| 888888885555588888888888888887777777 | |
| 888888885555588888888888888887777777 | |
| 888888885555588888888888888887777777 | |
| 888888885555588888888888888887777777 | |
| 888888885555588888888888888887777777 | |
| 888888885555577777776666663337777777 | |
| 888888885555577777776666663337777777 | |
| 888888885555577777776666663337777777 | |
| 888888885555577777776666663337777777 | |
| 888888885555577777776666663337777777 | |
| 888888885555577777776666663337777777 | |
| 888888885555577777778888888888888888 | |
| 888888885555577777778888888888888888 | |
| 888888885555577777778888888888888888 | |
| 888888885555577777778888888888888888 | |
| 444444445555577777778888888888888888 | |
| 444444445555577777778888888888888888 | |
| 444444445555577777778888888888888888 | |
| 444444445555577777778888888888888888 | |
| 0 | |
| N=9 -> 0 area omitted (alternate solution -- start is N=8 and add the 9s) | |
| 222233355555555555555566666666666666666655555 | |
| 222233355555555555555566666666666666666655555 | |
| 444433355555555555555566666666666666666655555 | |
| 444433355555555555555566666666666666666655555 | |
| 444433355555555555555566666666666666666655555 | |
| 444433377777778888888866666666666666666655555 | |
| 777777777777778888888877777777777777444455555 | |
| 777777777777778888888877777777777777444455555 | |
| 777777777777778888888877777777777777444455555 | |
| 777777777777778888888877777777777777444455555 | |
| 777777777777778888888877777777777777999999999 | |
| 777777777777778888888877777777777777999999999 | |
| 777777777777778888888877777777777777999999999 | |
| 777777777777776666668888888888888888999999999 | |
| 777777777777776666668888888888888888999999999 | |
| 777777777777776666668888888888888888999999999 | |
| 777777777777776666668888888888888888999999999 | |
| 777777777777776666668888888888888888999999999 | |
| 777777777777776666668888888888888888999999999 | |
| 777777714444888888888888888888888888999999999 | |
| 888888884444888888888888888888888888999999999 | |
| 888888884444888888889999999997777777999999999 | |
| 888888884444888888889999999997777777999999999 | |
| 888888884444888888889999999997777777999999999 | |
| 888888884444888888889999999997777777999999999 | |
| 888888884444888888889999999997777777999999999 | |
| 888888884444888888889999999997777777999999999 | |
| 888888889999999993339999999997777777999999999 | |
| 888888889999999993339999999998888888888888888 | |
| 888888889999999993339999999998888888888888888 | |
| 888888889999999996666666666668888888888888888 | |
| 888888889999999996666666666668888888888888888 | |
| 888888889999999996666666666668888888888888888 | |
| 888888889999999996666666666668888888888888888 | |
| 888888889999999996666666666668888888888888888 | |
| 888888889999999996666666666668888888888888888 | |
| 999999999999999999999999999999999999999999999 | |
| 999999999999999999999999999999999999999999999 | |
| 999999999999999999999999999999999999999999999 | |
| 999999999999999999999999999999999999999999999 | |
| 999999999999999999999999999999999999999999999 | |
| 999999999999999999999999999999999999999999999 | |
| 999999999999999999999999999999999999999999999 | |
| 999999999999999999999999999999999999999999999 | |
| 999999999999999999999999999999999999999999999 | |
| 0 | |
| N=10 -> 0 area omitted | |
| 2222333333999999999666666AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 2222333333999999999666666AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 4444333333999999999666666AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 4444666666999999999666666AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 4444666666999999999666666AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 4444666666999999999666666AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 4444666666999999999666666AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 4444666666999999999666666AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 4444666666999999999666666AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 4444555557777777333666666AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 4444555557777777333666666AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 4444555557777777333666666AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 4444555557777777999999999AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 4444555557777777999999999AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 9999999997777777999999999AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 9999999997777777999999999AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 9999999997777777999999999AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 9999999997777777999999999AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 9999999997777777999999999AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 9999999997777777999999999AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 9999999997777777999999999AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 9999999997777777999999999AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 9999999997777777999999999AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 8888888888888888999999999AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 8888888888888888999999999AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 8888888888888888999999999AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 8888888888888888999999999AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 8888888888888888999999999AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 8888888888888888999999999AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 8888888888888888999999999AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA | |
| 8888888888888888555558888888888888888999999999999999999 | |
| 8888888888888888555558888888888888888999999999999999999 | |
| 8888888888888888555558888888888888888999999999999999999 | |
| 8888888888888888555558888888888888888999999999999999999 | |
| 8888888888888888555558888888888888888999999999999999999 | |
| 8888888888888888555558888888888888888999999999999999999 | |
| 8888888888888888555558888888888888888999999999999999999 | |
| 8888888888888888555558888888888888888999999999999999999 | |
| 888888888888888855555666666AAAAAAAAAA999999999999999999 | |
| 888888887777777155555666666AAAAAAAAAA999999999999999999 | |
| 888888887777777666666666666AAAAAAAAAA999999999999999999 | |
| 888888887777777666666666666AAAAAAAAAA999999999999999999 | |
| 888888887777777666666666666AAAAAAAAAA999999999999999999 | |
| 888888887777777666666666666AAAAAAAAAA999999999999999999 | |
| 888888887777777666666666666AAAAAAAAAA999999999999999999 | |
| 888888887777777666666666666AAAAAAAAAA999999999999999999 | |
| 888888889999999994444666666AAAAAAAAAA999999999999999999 | |
| 888888889999999994444666666AAAAAAAAAA999999999999999999 | |
| 8888888899999999944446666667777777777777777777777777777 | |
| 8888888899999999944446666667777777777777777777777777777 | |
| 8888888899999999955555555557777777777777777777777777777 | |
| 8888888899999999955555555557777777777777777777777777777 | |
| 8888888899999999955555555557777777777777777777777777777 | |
| 8888888899999999955555555557777777777777777777777777777 | |
| 8888888899999999955555555557777777777777777777777777777 | |
| N=11 -> 0 area omitted (use solution for N=10, and just add 11s) | |
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