# Sunday Times Teaser 2996 – Piece It Together

*by Victor Bryant*

#### Published Sunday February 23 2020 (link)

I have some jigsaw-type pieces each consisting of one, two, three or four 1cm-by-1cm squares joined together without overlapping. The pieces are black on one side and white on the other, and they are all different. I have used all my pieces to simultaneously make some different-sized white squares in jigsaw fashion, with each square using more than one piece. Even if you knew what all my pieces were like, you would not be able to determine the sizes of my squares.

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This solution uses a collection of routines that I have put together in jigsaw.py

It uses either specific rectangle fitting code or Donald Knuth’s ‘Algorithm X’ (by changing ‘solve_rect as solve_rect’ to ‘solve_rectx as solve_rect’ in line 2). It produces the solution with the following two sets of squares in under a second.

The illustrations for the 4*4 and 5*5 have two pieces of 4 squares that are the same, Shouldn’t one of the four square pieces be three squares in a row with the fourth attached at the middle?

Hi Jim, He uses all his pieces and says that even if you knew what his pieces are you wouldn’t know the sizes of his squares. This means that he can make two different sets of squares with the same pieces, which is what the left and the right sides of the illustration show. The two sets of squares shown use the same nine pieces in total.

Understood but my issue is all the pieces must be different and in illustrations 16 and 25 there are two pieces in each that are identical.

The nine pieces ARE all different. And all nine can be used to build two different sets of squares as shown on the left and right above. There is NO requirement that the 16 and 25 squares do not share pieces (any more so than the two identical 4 squares!). The only requirements are (a) that the (nine) pieces are all different, (b) they can be used build two different sets of two or more squares using all nine of them, and (c) all the squares built must have two or more pieces.