Four of the five puzzle pieces shown can fit together without
overlap to form a square.
If the pieces can be rotated but not flipped,
which piece does not get used?
, 2026
Pick one
Solution
In this solution, we call the part of the puzzle piece that
sticks out, the tab. We call the recessed space, the
hole. A tab in one piece is designed to fit into a
corresponding hole in another piece.
Piece 1 has a tab in the shape of a trapezoid. This is the only piece
with a tab in this shape. Piece 4 and Piece 5 both have a hole in the
shape of a trapezoid. Since there is only one trapezoid tab, either
Piece 4 or Piece 5 will be the piece that does not get used to form a
square.
Piece 4 has a tab in the shape of a triangle. Piece 1 is the only
piece with a hole in the shape of a triangle. However, it is not
possible to insert the Piece 4 tab into the Piece 1 hole while also
inserting the Piece 1 tab into the Piece 4 hole. Thus, it is Piece 4
that does not get used to form the square. The completed puzzle is
shown.
Want a route through all this instead of an archive? The track
puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.