59. Drawing Gloves in the Dark A lady keeps gloves and hats together in her closet. Among the items there are exactly 14 blue gloves, 25 red gloves, and 45 yellow gloves (all other items are hats). Because it is completely dark she cannot see colors, but by touch she can always tell whether an item is a glove or a hat, and she removes an item only when she is sure it is a glove. What is the minimum number of gloves she must take out to be absolutely certain that she has at least two gloves of each color? Added 1 June 2011 Show Hint Show Solution Hint: Ask how many gloves she could take while still missing a pair in at least one color. Solution: To guarantee two gloves of every color, consider the worst-case draw. If she wishes to avoid getting two blue gloves, she could draw every red and yellow glove (25 + 45 = 70) and, at worst, just one blue glove. That is 70 + 1 = 71 gloves while still lacking a blue pair. Similarly, she could avoid a red pair with 14 + 45 + 1 = 60 gloves, or avoid a yellow pair with 14 + 25 + 1 = 40 gloves. The largest of these possibilities is 71. Therefore, after drawing 71 gloves it is still possible that she is missing a pair in one color (the blues). Consequently, as soon as she draws one more glove—making a total of 72—the worst case is eliminated and she is guaranteed to have at least two gloves of every color. Minimum required gloves: 72.
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