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Yesterday I've posted quite easy puzzle: Universal dissection. Now the actual problem.

When we deal with 8x8 board with 1 missing cell it doesn't matter whether we allow to flip parts or not, optimal number of parts will be the same (see answers of @elias and @Peter Taylor). I would like to have a similar puzzle, which would give different results when you allow and when you forbid fliping.

So how similar it should be? Here is the priorities, from high to low:

  1. The puzzle condition must look be like:

Alice has a piece of squared paper in shape-A (which covers integer number of cells). She cuts out shape-B from it, the shape-B center is placed at row N, column M, oriented as K. Bob cuts the rest of the paper into pieces. Once he is done, Alice asks Bob to put the pieces together in a way that they form shape-A paper with missing shape-B square at row X, column Y, orientation Z. What is the minimal number of pieces Bob must make to always be able to do what Alice says?

  1. Preferably shape-B should be a 1x1 cell, just like in the original puzzle.

  2. If not, then the shape-B should be as symetrical as possible (so orientation can be excluded from the puzzl condition) and as small as possible.

  3. Shape-A should be as simple as possible.

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To put up my opinion, that problem was kind of a regular competitional fake problem, replacing a not-so-trivial puzzle with a very special case of it, so now you can solve it instantly.

Consider this not-so-trivial version of the problem now, of which the fake gratification is removed:

From a complete symmetric tiling of the plane with hexagon, equilateral triangle and squares, we select a tile combination shape, which has more than two axis of symmetry. (Your problem is a very special case of this, where only squares are used, and there are four axis of symmetry, and the shape itself is a 8*8 square). Now A cuts out a triangles, b hexagons and c squares (Note that A cuts and keeps triangles from the shape ifff the symmetric periodic tiling contains triangles, same applies for hexagon and squares). Now B cuts the remaining shape, so that no matter from where A chooses a,b,c (The same numbers) triangle, hexagons and squares from the shape, B can tile the shape with the cut out pieces. For how minimum pieces, B can do it?

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