One might also consider dissections in which the pieces are not polyominoes; some of these might make better puzzles than polyomino dissections. Jim On Mon, Jan 9, 2017 at 7:31 PM, James Propp <jamespropp@gmail.com> wrote:
I hope you look for a minimal solution, Scott! (I wonder if anyone has already created software that automates the process?)
I should mention that I asked Matt Parker if any of his readers had invented dissection puzzles based on the identity (n+0)^2+(n+3)^2+(n+5)^2+(n+6)^2=(n+1)^2+(n+2)^2+(n+4)^2+(n+7)^2 for specific values of n and he said no.
In addition to n=0 and n=1 which I mentioned in my email, one might also consider negative values of n.
n=-1: 2^2 + 4^2 + 5^2 = 3^2 + 6^2 (I like this one) n=-2: 3^2 + 4^2 = 5^2 (hello, Pythagoras!) n=-3: 3^2 + 3^2 = 1^2 + 1^2 + 4^2 (not so interesting)
Here I've removed squares that occur on both sides of the equation. If one tries going beyond -3 to -4 etc., one finds that "reciprocity" sets in, and one doesn't get anything new.
By the way, partitioning the set {1,2,3,4,5,6,7} into {1,2,4,7} and {3,5,6} reminds me of the numbers 24 (hours in a day), 7 (days in a week), and 365 (days in a year). I'm not sure if there's any way to exploit this coincidence, though.
Jim
On Mon, Jan 9, 2017 at 6:55 PM, Scott Kim <scottekim1@gmail.com> wrote:
Wonderful questions. The second question lends itself to an especially nice presentation: use all these pieces to make exactly three squares. Then use the same pieces to make exactly four squares. The best version would have a unique solution for each of these challenges. Presentation for the first question is siimilar: use all these pieces to make exactly four squares. Now find a different solution. If no one has a dissection for these challenges, then I'm going to look for a minimal one.
On Mon, Jan 9, 2017 at 8:33 AM, James Propp <jamespropp@gmail.com> wrote:
Has anyone devised a fun puzzle in which a 1-by-1, 4-by-4, 6-by-6, and 7-by-7 square are divided into smaller polyominoes which can then be reassembled to form a 2-by-2, 3-by-3, 5-by-5, and 8-by-8 square?
(The title of the thread is a variation on Matt Parker's "Share the Power" puzzle, which is a generalization of this to higher powers, but without the embodiment via polyominoes.)
My guess is that the best puzzle of this kind (i.e., the most challenging to solve) would be one that used a near-minimal number of pieces.
I'm also seeking a polyomino implementation of the identity 0^2+3^2+5^2+6^2 = 1^2+2^2+4^2+7^2, though of course one of the squares has vanished!
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