Chris Maslanka devised this brainteaser for the Gathering for Gardner held in Atlanta in April 2004:
A bouquet contains red roses, whites roses, and blue roses. The total number of red roses and white roses is 100; the total number of white roses and blue roses is 53; and the total number of blue roses and red roses is less than that.
How many roses of each color are there?
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Call the number of red roses, white roses, and blue roses r, w, and b, respectively. So r + w = 100; w + b = 53; and b + r = x, where x is less than 53.
Adding these three equations gives 2r + 2b + 2w = 153 + x. The expression on the left is an even number, so 153 + x must be even as well. That means x is odd.
Since 2w + 2r = 200, we know that 200 + 2b = x + 153 and thus x = 47 + 2b. We know that the bouquet contains blue roses (that is, b is not zero) and that x is less than 53, so x is an odd number higher than 47 and lower than 53. So it’s 49 or 51.
If x = 49, then adding the three equations and dividing by 2 gives r + b + w = 101, and we have 1 blue rose, 52 white roses, and 48 red roses. But the problem states that the bouquet contains blue roses in the plural. So x must be 51, and we have 2 blue roses, 51 white roses, and 49 red roses.
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