In the problem "Mirror, mirror on the wall" it was proved that no number in the decimal system doubles on reversing its digits, and answered for bases 3, 5 and 8 (base 2 has leading zero, so itīs not valid).

Generalise the answer for positive integer bases.

With no proof given here, it seems base b, such that b modulo 3 is 0 will have a pattern as following, where each digit is represented as the calculation between the square brackets. The question mark (?) indicates a further inserted pattern can exist for other numbers in the same base, to include any number of [b-1] digits:

[b/3+1] [b/3] ? [2b/3-1] [2b/3] 
Its double: [2b/3] [2b/3-1] ? [b/3] [b/3+1]

For base b, such that b modulo 3 is 2, the following appears to be a pattern:

[(b+1)/3-1] ? [2(b+1)/3-1]
Its double: [2(b+1)/3-1] ? [(b+1)/3-1]

Posted by Dej Mar on 2008-09-03 01:00:41