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Problem of the Week
Problem B and Solution
Toothpicks for Squares

Problem

The diagram below is constructed from \(17\) toothpicks, creating a total of eight squares. Note that some of these are smaller squares of dimension \(1\) toothpick by \(1\) toothpick and some are larger squares of dimension \(2\) toothpicks by \(2\) toothpicks.

17 toothpicks are arranged to form a 2 by 3 grid. Three rows of three toothpicks placed horizontally end to end form the horizontal lines in the grid and four columns of two toothpicks placed vertically end to end form the vertical lines of the grid.

Start with the original diagram in each part below.

  1. Remove five toothpicks so that a total of five squares remain.

  2. Remove five toothpicks so that a total of three squares remain.

  3. Remove three toothpicks so that a total of two squares remain.

  4. Remove six toothpicks so that a total of two squares remain.

Compare your answers to those of a classmate. Are they the same? Can you complete each part without leaving extra toothpicks that do not belong to a square?

Solution

Answers will vary. The toothpicks removed are coloured white, and the toothpicks remaining coloured black.

  1. A solution with four small squares and one large square remaining is shown.

    The 5 toothpicks removed are the first toothpick in each of the horizontal lines and the two toothpicks in the first vertical line.

  2. A solution with three small squares remaining is shown.

    The 5 toothpicks removed are the first toothpick and last toothpick in the first horizontal line, the first toothpick in the first and last vertical lines, and the second toothpick in the last horizontal line.

  3. A solution with two large squares remaining is shown.

    The 3 toothpicks removed are the toothpicks in the middle horizontal line.

  4. A solution with one small and one large square remaining is shown.

    The 6 toothpicks removed are the first toothpick in the first horizontal line, the second and third toothpicks in the second horizontal line, the first toothpick in the first vertical line, and the two toothpicks in the third vertical line.