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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(2): Solution | Comment 4 of 7 |
(In reply to re: Solution by Charlie)

Comparing my computer results with the results of Brian's formula:

` k       computer          formula 1      0.2500000       0.250000000  2      0.0714286       0.035714286  3      0.0200000       0.003333333  4      0.0054945       0.000228938  5      0.0014881       0.000012401 `

the computer result seems to be k! times that of the formula indicating the formula should be

k!*(k+1)!/((2k)!*(3k+1))

I'll look over Brian's derivation to see if this makes sense in that context.

 Posted by Charlie on 2016-03-17 21:34:14

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