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This is probably a old puzzle which i encountered today:

A lady has fine gloves and hats in her closet- $18$ blue, $32$ red, and $25$ yellow. The lights are out and it is totally dark. In spite of the darkness, she can make out the difference between a hat and a glove. She takes out an item out of the closet only if she is sure that if it is a glove. How many (least) gloves must she take out to make sure she has a pair of each color?

There is two kinds of answers floating around in the internet $59$ or $60$.Could anybody explain how to solve this problem mathematically?

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2 Answers

up vote 3 down vote accepted

Assume that the narrative, which is unclear, means that there is at least one hat and one pair of gloves of each colour, that gloves come in pairs, and that the numbers given are totals of hats and gloves (taken singly). You get the maximum number of gloves required by first minimising the number of hats. Since there have to be an even number of gloves of each colour (they come in pairs) this gives hats: 2b, 2r, 1y; gloves: 16b, 30r, 24y.

Now, how many gloves can you take without getting a pair of each colour. That would be 30r + 24y + 8b (all left hands say) = 62. So on this interpretation you'd need to take 63 gloves.

Alternatively she may be able to tell the difference between a left-hand glove and a right-hand glove by touch. Then she would need to take 15r+12y+1b = 28 left-handed gloves to ensure a left-handed glove of each colour, and 28 right-handed gloves to ensure a right-handed glove of each colour - a total of 56.

So the answer depends on a number of assumptions which are unclear from the statement of the problem.

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Not much a combinatorics question though. Calculate the worst possible case that all the choices are from one group and then the next and so on. It's possible that all of first 32 can be red, and the next 25 can be yellow. Since she needs at least a pair of each, she needs 2 more to get a blue pair. Therefore, 32+25+2 = 59 is your answer.

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Not sure that this works - unsure what the number of gloves is - whether the numbers represent pairs or not, or whether they represent the total of hats and gloves, and whether the gloves are all in pairs. But gloves conventionally come in right-handed and left-handed versions, so she needs more than two blue gloves to guarantee a pair. –  Mark Bennet Aug 22 '11 at 20:03
    
@Mark Bennet:That's exactly my doubt. –  Quixotic Aug 23 '11 at 7:28
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