What happened to (0, Y, 1, X)?
It’s just a reflection of (0, X, 1, Y).
OK.
The “extra large” jumps occur earlier in the list as well. And, having allowed the program to run a bit longer, I can tell you that the solutions given up to 32 (1, 10610, 30123, 66013) are minimal solutions. There’s nothing like a bit of brute force.
I’ve verified that zut’s solutions up to 38 (1, 121416, 344733, 755477) are minimal. I don’t think I’ll do any more, it takes rather a lot of processor time.
Looks like zut nailed it. Kudos to him et al.
One other thing of interest: this same problem is being discussed here, and someone else pointed out that these sets of “squares” have a name: they’re called Ducci sequences. I gather a more general Ducci sequence has n numbers on each level (the one on this thread having n=4). There are one or two or three people that study these professionally.
Let’s do triangles.
I don’t understand half of it, but excellent links <B>zut.</B>
I can confirm those numbers up to n = 50. It looks to be correct.
What do you mean by “confirm those numbers”? That they are solutions, or that they are the smallest solutions?
Thanks to zut, and a big tip of the hat to Zenbeam for providing the analysis of the general case- great idea to reduce it to permutations of (0, X, Y, 1).
I was about to suggest extension to shapes with n sides- but I see I’ve been pre-empted!
They are the smallest.
A fairly simple problem, which turns out to be somewhat complex, which gets exhaustively examined unto death, … errr, I’ve gotten somewhat off the track.
Thanks to all who participated in a fascinating diversion.