Returning to Constraint 3, in the first case, digits 1 and 3 must be 1 and 7 in some order, and digits 7 and 9 must be 3 and 9 in some order (four possibilities: 147258369, 741258369, 147258963, 741258963)

In the second case, if 1 is in position 1 or 3, then 7 must be in position 7 or 9, and vice versa. Similarly with 3 and 9. We end up with 16 possibilities.

These constraints guarantee the number meeting all of the initial conditions except two. The first eight digits are only guaranteed to be divisible by 4, not by eight, and nothing is guaranteed concerning the first seven digits.

For the first eight digits to be divisble by 8, digits 7 and 8 must be 16, 32, 72, or 96.

This reduces the possibilities in the first case from four to two: (147258963 and 741258963), and in the second case, from sixteen to eight.

Posted by TomM on 2002-06-07 14:54:18