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Even though you're right in general, this is not the case here! We have proved that no smaller counter-example exists (again by using a computer search through

by bsubs 4y ago

Even though you're right in general, this is not the case here! We have proved that no smaller counter-example exists (again by using a computer search through optimized SAT-solving). As for the question of why no smaller counter-examples, I'm afraid I don't have any nice answers and perhaps there simply isn't a nice answer. Let me explain what I mean. My advisor Marijn Heule finished the resolution of Keller's conjecture (a conjecture about how N-dimensional cubes work in the Euclidean N-dimensional space), and the final answer is that the conjecture fails for the first time in dimension 8. "Why 8?" Again it's the same situation: it seems that that's just the way math is, something in the way the definitions and the objects behave makes it so that is the smallest counterexample. "Why" is a hard question to answer...

c7b 4y ago
Oh wow, cool work! Yeah, the 'why' question can be hard to answer. We already have a proof that spells out a perfectly rigorous answer, but often we're looking for a 'deeper' connection to something perceived as more fundamental.
- bsubs 4y ago
  I agree, and it would be amazing for someone to find a deeper intuitive reason for why 15 is the answer to this problem, or 8 is the answer to that other problem, or many such cases. This is however extremely hard in my opinion (note that I say "opinion" on purpose, because I don't have any way to formalize this idea). Let me use a couple more examples: any Rubik's cube (3x3x3) can be solved in at most 20 moves (on the standard metric, counting double turns as one move), and a Sudoku needs at least 17 clues to have a unique solution. Why 20? Why 17? We (humanity) got to both answers through computation, and even though the computation requires mathematical observations to be made more efficient, at the end we still checked an in-human number of cases... So in all honesty, if I had to bet, I'd say we will probably never intuitively know why 20 moves or why 17 clues, or in my case, why 15 colors... I'd love to be proven wrong tho!

by bsubs 4y ago

c7b 4y ago

Oh wow, cool work! Yeah, the 'why' question can be hard to answer. We already have a proof that spells out a perfectly rigorous answer, but often we're looking for a 'deeper' connection to something perceived as more fundamental.

bsubs 4y ago
I agree, and it would be amazing for someone to find a deeper intuitive reason for why 15 is the answer to this problem, or 8 is the answer to that other problem, or many such cases. This is however extremely hard in my opinion (note that I say "opinion" on purpose, because I don't have any way to formalize this idea). Let me use a couple more examples: any Rubik's cube (3x3x3) can be solved in at most 20 moves (on the standard metric, counting double turns as one move), and a Sudoku needs at least 17 clues to have a unique solution. Why 20? Why 17? We (humanity) got to both answers through computation, and even though the computation requires mathematical observations to be made more efficient, at the end we still checked an in-human number of cases... So in all honesty, if I had to bet, I'd say we will probably never intuitively know why 20 moves or why 17 clues, or in my case, why 15 colors... I'd love to be proven wrong tho!