Problem:
Eight square tiles are arranged as shown so their outside edges form a polygon with a perimeter of units. Two additional tiles of the same size are added to the figure so that at least one side of each tile is shared with a side of one of the squares in the original figure. Which of the following could be the perimeter of the new figure?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
When a new tile is added to the original figure, it may have one or two sides in common with the given tiles, as shown. When a tile shares one side, the original perimeter is increased by . When a tile shares two sides, there is no change in the perimeter. By adding two tiles, the only possible changes to the perimeter are increases of or . Hence, the possible values of the perimeter are or .
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Note. Examples of the three possibilities are shown.
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Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions