My drawer has n pairs of socks (so 2n socks). The socks came backfrom the laundry, surprisingly with no sock missing! But they are allseparated. Each sock is labeled either “left” or “right”. Assume thatI grab some socks blindly, how many do I need to guarantee a goodpair?• If I must use a correct pair of socks (two that match). Show howto use the concept of pigeonhole to find the answer.• If I am happy to use any left sock with any right sock (this is notpigeonhole, but it has some logic to it).
Question
My drawer has n pairs of socks (so 2n socks). The socks came backfrom the laundry, surprisingly with no sock missing! But they are allseparated. Each sock is labeled either “left” or “right”. Assume thatI grab some socks blindly, how many do I need to guarantee a goodpair?• If I must use a correct pair of socks (two that match). Show howto use the concept of pigeonhole to find the answer.• If I am happy to use any left sock with any right sock (this is notpigeonhole, but it has some logic to it).
Solution
To guarantee a good pair of socks, you need to consider two scenarios:
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If you must use a correct pair of socks (two that match), you would need to grab n+1 socks to guarantee a pair. This is because if you grab n socks, you could potentially grab all left socks or all right socks. By grabbing one more (n+1), you are guaranteed to have at least one pair that matches. This is an application of the Pigeonhole Principle, which states that if you have more items (socks) than categories (pairs), at least one category must have more than one item.
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If you are happy to use any left sock with any right sock, you would need to grab just 2 socks to guarantee a pair. This is because you only need one left sock and one right sock to make a pair, regardless of whether they match or not. This is not an application of the Pigeonhole Principle, but rather simple logic.
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