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Sum of three adjacent numbers on a wheel
By Anonymous on Thursday, September 28, 2000 - 08:27 am:

Hi!

I am stuck on the following problem:

A wheel has integers 1-25 placed on it in a random manner. Show that there must be 3 adjacent numbers whose sum is at least 39.

I don't really know how to start this problem!
I would appreciate if you could help in any way!

Thank you very much

By Arvan Pritchard (T708) on Thursday, September 28, 2000 - 10:10 am:

You could start by looking at the average value of all the sums of 3 adjacent numbers (it's going to be 3 times the average value of a number - can you see why?).

Then all the sums are equal to this or some are higher and some lower (because it's the average).

By Marcus Hill (T3280) on Thursday, September 28, 2000 - 12:07 pm:

And can you see why the numbers need to be on a wheel rather than in a line?

By Anonymous on Thursday, September 28, 2000 - 01:51 pm:

So,

let m = mean value of number = 13

let E(X) = expected value of sums of 3 adjacent numbers = 3m = 39

Thus, since it's an average, there must be 3 adjacent numbers whose sum is at least 39 (the average)!

I don't quite see why the numbers must be on a circle/wheel but thanks for pointing that out! I'll try to figure it out!

Thank you very much!