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What is the number of lines that pass through the center of an n-dimensional tic-tac-toe grid?

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This question is at too low a level for MO (which is intended for research level questions) and poorly stated (what lines are you talking about? Infinitely many lines pass through every point in n-space...). See the FAQ for a list of places to ask elementary questions. I've voted to close. –  Andy Putman Aug 21 '10 at 19:18
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I planned to post it in math.statexchange.com, but my account was inoperable for 5 hours since I joined. I posted a query here regarding that (since I could not even access meta) and it was promptly deleted. See here: meta.math.stackexchange.com/questions/697/… –  user8682 Aug 21 '10 at 20:24
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closed as too localized by Robin Chapman, Andy Putman, Charles Siegel, François G. Dorais Aug 21 '10 at 19:38

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1 Answer

up vote 2 down vote accepted

The squares tic tac toe grid can be represented with $(x^1,x^2,...,x^n)$ where each can be 0, 1 or 2.

A line passing through the center can be identified Given any coordinate except $(0,0,...,0)$.

There are $3^n-1$ possible coordinates to start a line from.

For each line through center, there will be two endpoints. This should help you get your answer.

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The way this is written makes it seem like you're suggesting the answer is 3^n-1 choose 2. (If this isn't the case, please ignore this comment!) But this is much bigger than the correct answer. The reasons for this are that (1) not every choice of two endpoints gives a line through the centre, and (2) for each line, there are several choices of a pair of endpoints which yield that line. –  Brad Hannigan-Daley Aug 21 '10 at 21:54
    
Brad, he means (1,1,...,1) instead of (0,0,...,0), that is, the center itself, and then consider the line through any endpoint through the center, counted twice, etc. –  Joel David Hamkins Aug 22 '10 at 0:31
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