This question on MathOverflow discusses the problem of finding the distribution 
of the maximum value of a multinomially-distributed random variable: 
https://mathoverflow.net/questions/104948/distribution-of-maximum-of-a-uniform-multinomial-distribution

Using the formula on that page, and the bounds on test set 1, I get an upper 
bound of P(max >= 18) <= 0.8131. 

This drastically overestimates the actual probability, but it's still 
significantly less than 1. So we can perform multiple trials, and reduce the 
probability of getting incorrect results on _all_ of them to a very small 
number. (With 365 trials, the chance of error is at most 0.8131^365 ~= 10^-33.)

-- David

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