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 Never 0 mod 3 (Posted on 2016-03-17)
Let k be a positive integer. Suppose that the integers 1, 2, 3, ...3k, 3k + 1 are written down in random order.

What is the probability that at no time during this process, the sum of the integers that have been written up to that time is divisible by 3?

Source: Putnam competition

 No Solution Yet Submitted by Ady TZIDON No Rating

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 re: Solution | Comment 3 of 7 |
(In reply to Solution by Brian Smith)

These are the values I get using Brian's formula:

1                      0.250000000
2                      0.035714286
3                      0.003333333
4                      0.000228938
5                      0.000012401

for the first five values of k, which disagrees with the direct trials of all possibilities mod 3 in my previous post.

I repeated the case of k=2, using "1234567" instead of "1231231" and got the same results as before: 0.0714286.

 Posted by Charlie on 2016-03-17 14:47:17

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