This brainteaser was written by Derrick Niederman. A 10 × 10 grid is painted with three primary colors (red, yellow, and blue) and three secondary colors (green, purple, and orange). The secondary colors are made by mixing equal parts of the appropriate primary colors — that is, red and yellow are mixed to make orange, red and blue to make purple, and yellow and blue to make green. The figure below shows squares that were painted red and blue. No other squares were painted either red or blue. Suppose that each small square requires a quart of paint. Altogether, 31 quarts of red paint, 40 quarts of blue paint, and 29 quarts of yellow paint were used to paint the entire 10 × 10 grid. Given this information, can you determine if there were more yellow or purple squares? And how many more? Resources for Teaching Math
© 2009 National Council of Teachers of Mathematics, Inc.
http://illuminations.nctm.org
Solution: purple; 8 more purple squares than yellow. One possible arrangement of the colors is shown below. Note that the number of purple squares exceeds the number of yellow squares by eight. For a general solution, let p represent the number of purple squares. Then p/2 quarts of red paint were used to make purple. But since there were 31 quarts of red to start with and 14 squares were painted red, then it must be true that 31 – 14 – p/2 = 17 – p/2 quarts of red were used to make orange paint. Consequently, an equal amount of yellow must have been used to make the orange paint, so there must be 2 × (17 – p/2) = 34 – p orange squares. Similar reasoning says that p/2 quarts of blue paint must have been used for the purple, so it must also be the case that the remaining 40 – 20 – p/2 = 20 – p/2 quarts of blue were used to make green. Hence, 2 × (20 – p/2) = 40 – p squares must be green. So adding all of the amounts together gives: 14 + 20 + p + (34 – p) + (40 – p) + y = 100 red blue purple orange green yellow Simplifying gives ‐p + y = ‐8, or p – y = 8. Consequently, any arrangement will have eight more purple squares than yellow squares. In the diagram above, there are 23 purple squares and 15 yellow squares, but many other arrangements with p – y = 8 are also possible. Resources for Teaching Math
© 2009 National Council of Teachers of Mathematics, Inc.
http://illuminations.nctm.org