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I came across this puzzle at some online contest(now completed)

Alexander, son of Phillip of Macedon, has ascended the throne of his father following his assassination. In these tumultuous times, he appoints you as head of the architectural division of his mighty army. The general of the army wants all the catapults to be inducted in the artillery which have a n-sided polygon base, and no two or more of them should have the same type of polygon as their base.However, for the structures to be agile and economic, it is required that the n should be an odd number and the polygon of the base should be constructable with help of a compass and straightedge.

Given the above situation, you are required to find out the total number of logs of woods required - each log of wood for each side of the base of structure - for the construction of all such catapults

I thought the result would be equal to the sum of the first 5 Fermat's Prime but unfortunately that wasn't correct.

Do you think the data provided is not sufficient or am I missing something? What could the possible answer to this question?

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    The Fermat primes aren't the only values of n for which n-gons are constructible; they're the only _prime_ values of n with this property. Read further in the Wikipedia article. (And where does it say in the problem that there are 5 catapults?)2011-01-27
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    Well n should be a product of 2^ and Fermat's prime but that is out of my reach. `5` was my wild guess.2011-01-27
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    The answer is 7304603327.2011-01-27
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    @Sven : How? Can you please explain?2011-01-27
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    It is not known whether there are infinitely many Fermat primes, so it seems to me that the answer to this puzzle depends on an unsolved problem in number theory...2011-01-27
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    @RahulNarain You could make an argument that any Fermat primes larger than the ones we know don't actually lead to polygons 'constructible with help of compass and straightedge' because the values involved would require a subatomic compass to resolve? :-)2013-06-18
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    Actually, there's a flaw in this problem, since it seems to assume that we know every fermat prime and, last time I checked it, we don't.2013-06-18

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You're missing something. Read what your Wikipedia link has to say about constructible polygons again:

An n-sided regular polygon can be constructed with compass and straightedge if and only if n is the product of a power of 2 and distinct Fermat primes. In other words, if and only if n is of the form $n = 2^kp_1p_2…p_s$, where k is a nonnegative integer and the $p_i$ are distinct Fermat primes.

(My emphasis)

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    What am I missing? Please do specify?2011-01-27
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    http://en.wikipedia.org/wiki/Fermat_number#Relationship_to_constructible_polygons It's only two sentences.2011-01-27
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A fairly specific hint: is a 15-gon constructable or not? How, and why?

(Hint to the hint: what is $\dfrac{2\pi}{5}-\dfrac{\pi}{3}$? Why does this matter?)

Now can you see which $n$-gons you might have left off your list?