In this case, I am assuming that each street and avenue must form a block, so the only case is where the two blocks are adjacent. Therefore, the total distance of the streets and avenues is 7, but there are 26 phone booths. Through pigeonhole, we know that some have to be less than 1 mi apart. Is this answer correct?
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Do you mean less than one block apart? I do not understand the reference to $1$ mi. – N. F. Taussig May 24 '17 at 00:03
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It means that the booths cannot be more than one mile, so a booth could sit in the middle of a block and another one would sit in the middle of the next block, so they would be (at most) one mile apart. – Gerard L. May 24 '17 at 00:05
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I would think that there are 16 blocks (in a 4-by-4 grid), and the total length of streets and avenues is 40 miles. – Per Erik Manne May 24 '17 at 00:11
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Then there is a hidden assumption that each block is one mile long. Notice that there are $25$ corners. If each phone booth could be placed on a corner, the shortest distance between them would be $1$ block. However, you have $26$ phone booths, so ... – N. F. Taussig May 24 '17 at 00:12
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As mentioned in the comments, there are 25 corners, where a street meets an avenue. Each booth has a nearest corner (or more than one, if it is exactly halfway between two adjacent corners of the same block), at most half a mile away. Since there are 26 booths, at least two of them must have the same nearest corner.
Per Erik Manne
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