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    <p>I would approach the problem (naively, to be sure) by taking a pessimistic view. I'd try to prove there was no 5x5 solution, and therefore certainly not <em>four</em> 5x5 solutions. To prove there was no 5x5 solution I'd try to construct one from all possibilities. If my conjecture failed and I was able to construct a 5x5 solution, well, then, I'd have a way to construct 5x5 solutions and I would try to construct <em>all</em> of the (independent) 5x5 solutions. If there were at least 4, then I would determine if some combination satisfied the letter count restrictions.</p> <p>[Edit] Null Set has determined that there are "4,430,974 5x5 solutions". Are these valid? I mean that we have a limitation on the number of letters we can use. This limitation can be expressed as a boundary vector BV = [9, 2, 2, 4, ...] corresponding to the limits on A, B, C, etc. (You see this vector in Null Set's code). A 5x5 solution is valid if each term of its letter count vector is less than the corresponding term in BV. It would be easy to check if a 5x5 solution is valid as it was created. Perhaps the 4,430,974 number can be reduced, say to N.</p> <p>Regardless, we can state the problem as: find four letter count vectors among the N whose sum is equal to BV. There are (N, 4) possible sums ("N choose 4"). With N equal to 4 million this is still on the order of 10^25---not an encouraging number. Perhaps you could search for four whose first terms sum to 9, and if so checking that their second terms sum to 2, etc. </p> <p>I'd remark that after choosing 4 from N the computations are independent, so if you have a multi-core machine you can make this go faster with a parallel solution. </p> <p>[Edit2] Parallelizing probably wouldn't make much difference, though. At this point I might take an optimistic view: there are certainly more 5x5 solutions than I expected, so there may be more final solutions than expected, too. Perhaps you might not have to get far into the 10^25 to hit one.</p>
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    1. COConstructing them naively is going to take O(26^25) operations. That's a massive number and far far too slow.
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    2. COThere certainly is a 5x5 solution printed in Knuth volume 4A -- more than one. He uses a different, smaller dictionary and doesn't limit the tiles available -- so if there's no solution, it's probably because of the tile set rather than the dictionary. I'd expect there to still be solutions, though.
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    3. CO@Darius I just implemented my solution and holy crap there are a TON of 5x5 word blocks. I stopped printing them when my output file was twice the size of the whole SOWPODS file.
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