A problem in May Lunchtime 2014, Codechef.com

So i was trying to solve this problem.

 You are given a multiset of N integers. Please find such a nonempty subset of it that the sum of the subset's elements is divisible by N. Otherwise, state that this subset doesn't exist.

What i did was simply generate the combinations and check if each of them was divisible by its cardinality. This solution turned out just fine.

After the competetion i found an elegant solution here.

It so turns out that i cannot generate a sequence of n numbers without having a consecutive subset which is not a multiple of n.

That is to say, whatever combination of numbers i try..there always exists a consecutive subset which is a multiple of 'No. of numbers'(n).

Can anyone explain this theoritically? Or is there any theorem that states this?

PS: I already tried Googling it...all in vain.

Note by Varshith Reddy
7 years, 2 months ago

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1 vote

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This is a classic application of Pigeonhole. Let the terms be a1a_1, a2a_2, \dots, ana_n. Consider the nn sums s1=a1,s2=a1+a2,,sn=a1+a2++an. \begin{aligned} s_1 &= a_1, \\ s_2 &= a_1 + a_2, \\ &\dots, \\ s_n &= a_1 + a_2 + \dots + a_n. \end{aligned} If any of these sums are divisible by nn, then we are done.

Otherwise, each of these sums are congruent to one of 1, 2, \dots, n1n - 1 modulo nn. There are nn sums and n1n - 1 possible residues modulo nn, so by Pigeonhole, two residues must be the same. In other words, there exist ii and jj, i<ji < j, such that sisj(modn)s_i \equiv s_j \pmod{n}. Then sjsi=ai+1+ai+2++ajs_j - s_i = a_{i + 1} + a_{i + 2} + \dots + a_j is divisible by nn, as desired.

Furthermore, the example a1=a2==an1=1a_1 = a_2 = \dots = a_{n - 1} = 1 shows that you need at least nn terms to guarantee that you can find a sum of consecutive terms that are divisible by nn.

Jon Haussmann - 7 years, 2 months ago

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Oh. I first though to apply Pigeonhole to a different part of the problem. @Sreejato Bhattacharya remember that Pigeonhole trick/thing you showed me? By the way I'm totally showing that to the class. :D

Finn Hulse - 7 years, 2 months ago

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Cool! :)

Sreejato Bhattacharya - 7 years, 2 months ago

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shaan ragib - 7 years, 1 month ago

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Interesting problem. Possibly a Pigeonhole argument is needed?

Calvin Lin Staff - 7 years, 2 months ago

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not a question

Rohan Lad - 7 years, 1 month ago

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