Can you admit you were wrong yet? Unless the number of kids evenly divides 60, then some kids will have better odds than others. Just some small admission that the odds don't work out perfectly unless N is 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, or 60.

Edit: For instance, if there are 59 kids, the first one has twice as good a chance to win as any of the others, because both 0 % 59 and 59 % 59 are 0. In other words, in the set derived from x % 59, for x in {0, 1, ..., 59}, 0 appears twice, and each other number in the range 0-59 only appears once.