Hi

Mathimagics,

Last question first: The solution grid has 12 automorphisms, and so the canonicalization routine tried 11 other transformations on the puzzle, looking for a smaller minlex puzzle to be produced. It found at least one, it seems.

blue wrote:For all but ~460000 grids (the ones that I knew had 12's, and the grids with non-trivial automorphisms) ...

Do you mean the grids that you knew had 12-clue solutions? But that's only 45K, if I understand correctly. Are the others all non-trivial automorphism cases? In which case why does that influence the HSet enumeration outcome?

Yes the others were cases with non-trivial automorphisms.

There are 414489 such grids, in total, or ~0.772% of the total.

There was an overlap between the two grid sets.

355 of the 45093 grids with known 12's, also had non-trivial automorphisms ... ~0.787% of them ... not unexpected, I guess.

Early on, seeing that 3 of the 12 grids with (known) 11's, had >1 automorphism, I was interested to know whether any of the other such grids, also had 11's. Looking into that, I ran into a lot of "killer" testing times, it seemed. When I tested random grids, though, that didn't seem to happen ... or at least not very often. That's why I separated them out, along with the grids with 12's, which I naturally assumed would take longer than the others.

About the effect on hitting set counts: In the end, I think that except for the "killer time" cases, the grids with >1 automorphism, actually tested a little faster than the average. On the other hand, after seeing your post, I've gone back and looked at the automorphism counts for the worst case grids. I had 12 grids that took > 1000 seconds. All but one of them has non-trivial automorphisms. The six grids from your earlier post are included in the list, and they all have >1 automorphism.

- Code: Select all
` grid # | #Aut | timing | HS count`

-----------+------+-------------+-------------

# 53666688 | 108 | 25915.0625s | 20047963823

# 23199970 | 54 | 14386.8906s | 47561944352

# 53658890 | 2 | 8654.9219s | 17877667846 (1 ED 11, 2 total)

# 53666687 | 1296 | 7512.3438s | 8877298313

# 53664625 | 4 | 4898.0781s | 10456885720

# 53658889 | 2 | 5226.7656s | 11898451461 (1 ED 11, 2 total)

# 23199968 | 4 | 4367.4688s | 9295824254

# 53658888 | 1 | 2578.0781s | 7458593883

# 23199962 | 2 | 2562.9531s | 7383782810

# 53577383 | 6 | 1766.5938s | 3626494540

# 53577379 | 2 | 1722.4063s | 3099601071

# 23199965 | 2 | 1186.8125s | 2461006110

123456789564897231978312645647938512389125476251764893895271364712643958436589127 # 53666688

123456789456897231978312456564978312789123564231564897897231645312645978645789123 # 23199970

123456789457891236968372451571948362689723145234615897895237614312564978746189523 # 53658890

123456789564897231978312645645978312789123456231564897897231564312645978456789123 # 53666687

123456789457918362896237154671894235589723641342561978968342517234175896715689423 # 53664625

123456789457891236968372451571948362689523174234617895895234617312765948746189523 # 53658889

123456789456897231978312456247968315589173624631524897894631572312745968765289143 # 23199968

123456789457891236968372451571938642389624175246715893895267314712543968634189527 # 53658888

123456789456897231978312456245968317789143625631725894894631572312574968567289143 # 23199962

123456789457298613869371452594837261382169574716524938678912345931745826245683197 # 53577383

123456789457298613869371452578912364931764825246583197695837241382149576714625938 # 53577379

123456789456897231978312456247638915389125674561749823894571362712963548635284197 # 23199965

I don't have a good explanation for why things should be like that.

There's this: for the top time grid, with 108 automorpihsms ... if U is an unavoidable set, then there are up to 108 distinct versions of U, obtained through transformation by one of the automorphisms. If U was a large 2-digit set, that would suggest that there are a lot of large 2-digit sets in the grid -- leaving "not much room" for smaller 2-digit sets. The same thing works the other way too, though -- if U was small, there should be a lot of small UA sets.

That all makes some kind of sense, but then most of the grids listed above, have relatively small automorphism counts. In the end, it's all a mystery to me.

Cheers,

Blue.