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Later addition Under more relaxed conditions, here is an example with 32 pieces (and a suggestion that maybe there is no limit) . If you look at it the right way you might be able to convince yourself that each piece is piecewise path-wise connected. The 4 colored pieces are congruent by rotation around the center of the area shown (which is a quarter of the full square). Using a reflection will give 4 more pieces for a total of eight. These eight pieces (so far) fit together to fill in the square shown and each touches each corner. Put together 4 copies of this (32 pieces total) to get the full square.

As it is, each piece in the full square touches one corner and the center. I feel have not totally convinced myself, but it seems that I it should be able to figure out if it is possible to divide each piece into 4 by quadrisecting each acute angle and then recolor in such a way as to have 64 or even more all 32 pieces each touching all 4 corners. If so, but I haven't yetthen 4 copies of that figure could be arranged and give a square partitioned into 132 pieces all touching its center. If that is correct then there should be no limit.

1) If you want respect on this site then stop hinting that you have a perhaps complete classification with three families plus one sporadic (I don't mind that but some people here do). Describe them carefully enough that people can decide if they have others that they can think of. Is the sporadic case the entire square? I gather that you think every solution uses 1,2,4 or 8 pieces. I imagine that is true but the proof of that alone would be a good start and might not be that easy (see comment 3 below.)

2) Your description would probably make clear what you mean by piece, but the most general definition commonly seen (although I have now used a looser definition above) might be something like : "a closed topological disk in the plane with boundary a simple closed curve." You wish to find a finite set of such pieces, all congruent (reflection allowed), disjoint interiors, union the square and the center on the boundary of each piece (ignoring the one piece case...). Tedious, but worth saying anyway. For reference below: a polyomino is such a tile made of unit squares meeting edge to edge.

Iit It might be relevant for this problem and at least is an example of how to prove such things.

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Later addition Under more relaxed conditions, here is an example with 32 pieces. If you look at it the right way you might be able to convince yourself that each is piecewise connected. The 4 colored pieces are congruent by rotation around the center of the area shown (which is a quarter of the full square). Using a reflection will give 4 more pieces for a total of eight. These eight pieces (so far) fit together to fill in the square shown and each touches each corner. Put together 4 copies of this (32 pieces total) to get the full square. I feel that I should be able to figure out if it is possible to have 64 or even more pieces, but I haven't yet.

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1) If you want respect on this site then stop hinting that you have a perhaps complete classification with three families plus one sporadic. Describe them carefully enough that people can decide if they have others that they can think of. Is the sporadic case the entire square? I gather that you think every solution uses 1,2,4 or 8 pieces. I imagine that is true but the proof of that alone would be a good start and might not be that easy (see comment 3 below.)

2) Your description would probably make clear what you mean by piece, but the most general definition commonly seen might be something like : "a closed topological disk in the plane with boundary a simple closed curve." You wish to find a finite set of such pieces, all congruent (reflection allowed), disjoint interiors, union the square and the center on the boundary of each piece (ignoring the one piece case...). Tedious, but worth saying anyway. For reference below: a polyomino is such a tile made of unit squares meeting edge to edge.

3) It is fun and challenging to find a missed example for claims of the sort: "this is all tilings." It can be surprisingly difficult (compared to the "obviousness" of the result) and rather tedious (to my taste) to prove that there are not any exceptions. Here is a 7 page paper discussing when you can split a polygon into 2 congruent shapes:

Splitting a Polygon into Two Congruent PiecesKimmo Eriksson The American Mathematical Monthly Vol. 103, No. 5 (May, 1996), pp. 393-400 Iit might be relevant for this problem and at least is an example of how to prove such things.

I can believe that the only way to partition a rectangle into 3 congruent pieces is if the pieces are themselves rectangles. Here is an 8 page proof:

Samuel J. Maltby Trisecting a rectangle Journal of Combinatorial Theory, Series A Volume 66, Issue 1, April 1994, Pages 40-52

It cites the following classic (to those who follow these matters) 6 page note proving a long conjectured result (again the methods in the paper could be relevant):

Polyominoes of order 3 do not exist I. N. Stewart and A. Wormstein Journal of Combinatorial Theory, Series A Volume 61, Issue 1, September 1992, Pages 130-136

Abstract: The order of a polyomino is the minimum number of congruent copies that can tile a rectangle. It is an open question whether any polyomino can have an odd order greater than one. Klarner has conjectured that no polyomino of order three exists. We prove Klarner's conjecture by showing that if three congruent copies of a polyomino tile a rectangle then the polyomino itself is rectangular. The proof uses simple observations about the topology of a hypothetical tiling, and symmetry arguments play a key role.

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