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Combinatorics Difficulty 2.6 Junior Find the answer Canada

Four of the five puzzle pieces shown can fit together without
overlap to form a square.

Hide/Reveal Description of Diagram for Question 17

The puzzle pieces are made from identical square shapes. Each top side has a shape that sticks out (a tab) and each right side has a shape that is cut out (a hole). Each tab and each hole has one of the following shapes: trapezoid, semicircle, or triangle. Each tab of a particular shape is designed to fit into the hole of the same shape. The tabs and holes on the pieces are as follows:

Piece 1: trapezoid tab, triangle hole
Piece 2: semicircle tab, semicircle hole
Piece 3: triangle tab, semicircle hole
Piece 4: triangle tab, trapezoid hole
Piece 5: semicircle tab, trapezoid hole

If the pieces can be rotated but not flipped,
which piece does not get used?

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.

Figure for this problem

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.