Analytic solution, taking Ady's hint and running with it.
Every 6 digit number meeting the first 2 criteria could be scaled by adding or subtracting the same amount from each digit to make the smallest digit equal to 1. This isn't necessary but helps me to visualize it. All these numbers will be some permutation of 123456.
To be divisible by 11, the even digits minus the odd digits must be 11,0, or -11.
To prove that the cardinality of S3 is zero, all that is necessary is to show this can never be true for any permutation of 123456.
A permutation that maximizes that function is 142536. (4+5+6) - (1+2+3) = 9, so there is no way to get to 11 or -11.
The sum of digits of 123456 is 21, so there is no way the function can be zero since there is no 3 digit subset of the 6 digits such that the sum of its digits would be 10.5.
Posted by Larry
on 2020-07-07 08:35:51