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This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with plain old algebra, which yields the shortest, simplest proofs, but other than that, the textbook gave no hints really and I'm really not sure about how to approach it. Any guidance hints or help would be truly greatly appreciated. Thanks in advance :) So anyway, here the problem goes:

The numbers $1-12$ are to be placed around a circle, as on a clock, but in any order. Show that there are three consecutive numbers in the arrangement with a sum of at least $19$.

MickG
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Taken directly from http://gottfriedville.net/mathprob/nt-circle12.html

Proof #1:

The sum of the $12$ numbers is $78$. If we add up all the sums of three adjacent numbers, we use each number three times, for a total of $234$. If all the sums are to be $< 19$, then their maximum total would be $12 \times 18 = 216$, which is too small.

In fact, there must be a sum of at least $20$, since $12 \times 19 = 228$, which is still too small.

Proof #2:

To avoid a sum of at least $19$, the numbers $12, 11, 10$, and $9$ must all be separated by at least two other numbers. Otherwise, two of them, plus any of the others totals at least $20$. Let the $12$ places be as follows: _ $X$ _ _ $X$ _ _ $9$ _ _ $X$ _ where $X$ indicates $10, 11$, and $12$ in some order. Since each of the dashes is part of some $3$-element sum with one of the $X$'s, none of them can accommodate $8$, without creating a sum that is at least $8 + 10 + 1 = 19$.

Cookie
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anonymous
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    Copied directly from http://www.gottfriedville.net/mathprob/nt-circle12.html – miradulo Jun 20 '15 at 01:17
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    @DonkeyKong Here I was praising the speed at which this was typed.. I feel lied to. – Cameron Williams Jun 20 '15 at 01:19
  • @CameronWilliams Nevertheless, the answer is correct. – anonymous Jun 20 '15 at 01:20
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    @CameronWilliams I had the same feeling, a proof a minute is impressive. Copied it into Google and what do you know. – miradulo Jun 20 '15 at 01:20
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    @anonymous It is someone else's answer, with no credit given to who actually wrote it. You can take pride in your Googling though. – miradulo Jun 20 '15 at 01:21
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    @DonkeyKong Wow, thanks for the link. –  Jun 20 '15 at 01:21
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    @anonymous It is definitely correct. I was just about to sing your praises for typing such a clear answer in such a short period of time. You should reference your sources in the future :) Nothing wrong with that. You're presumably an academic. You'll have to do that plenty in the future. – Cameron Williams Jun 20 '15 at 01:22
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    Wow, what a fast impressive answer @anonymous –  Jun 20 '15 at 01:22
  • @DonkeyKong I am new to MSE, but I am a math student graduate from New York University. I simply found a suitable answer. Cited sources, please do not continue to down vote. – anonymous Jun 20 '15 at 01:24
  • @anonymous I think in general, copying off sources found Googling is never a good idea. Anyways, I appreciate you changing it. I didn't downvote, not enough rep! – miradulo Jun 20 '15 at 01:29
  • @DonkeyKong Oh dear, I suppose I will resort to original answers then, although MSE is a highly competitive forum. I praise you for not downvoting! – anonymous Jun 20 '15 at 01:31
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    @DonkeyKong You have enough rep now. :-) – Cookie Jun 20 '15 at 01:34
  • @lesguimauves Oh dear, upvoting signifies the answer is useful. Surely my answer is useful? I am fearful of this encouragement of down voting destroying my ethic. – anonymous Jun 20 '15 at 01:36
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    @anonymous My view is that answers on Stack Exchange sites don't always have to be completely original work, just that they answer the OP's question. Besides, you attributed the original source, so you should be all good. In fact, I liked this answer you posted (I upvoted you), and apparently so does the OP since he/she accepted your answer. – Cookie Jun 20 '15 at 01:46
  • @lesguimauves Thank you. – anonymous Jun 20 '15 at 01:52