Problem:
Every card in a deck has a picture of one shape - circle, square, or triangle, which is painted in one of three colors - red, blue, or green. Furthermore, each color is applied in one of three shades - light, medium, or dark. The deck has cards, with every shape-color-shade combination represented. A set of three cards from the deck is called complementary if all of the following statements are true:
i. Either each of the three cards has a different shape or all three of the cards have the same shape.
ii. Either each of the three cards has a different color or all three of the cards have the same color.
iii. Either each of the three cards has a different shade or all three of the cards have the same shade.
How many different complementary three-card sets are there?
Solution:
Consider any pair of cards from the deck. We show that there is exactly one card that can be added to this pair to make a complementary set. If the cards in the pair have the same shape, then the third card must also have this shape, while if the cards have different shapes, then the third card must have the one shape that differs from them. In either case, the shape on the third card is uniquely determined. Similar reasoning shows that the color and the shade on the third card are also uniquely determined.Thus we can count the number of complementary sets by counting the number of pairs of cards and then dividing by , because each complementary set is counted three times by this procedure. The number of complementary sets is
The problems on this page are the property of the MAA's American Mathematics Competitions