Hello Math-Funsters, Take any positive integer and transform it via two successive operations – first reduction, then amplification. a) Reduction: all strings of same-parity digits are replaced par the sum of the said digits. Iterate the reduction if necessary. Example with 156186: 156186 – 66114 66114 – 1224 1224 – 18 b) Amplification: when an integer can no longer be reduced, replace every digit of it by its square (and concatenate the results). Thus 18 becomes: 18 – 164 Iterating the whole procedure gives (for 156186): 156186 – 18 – 2 – 4 – 16 – 136 – 10 – 10 – 10... [loop] Is there a positive integer somewhere which never loops? Is there a finite quantity of loops? 125 doesn’t end in the 10-loop: 125 – 1425 165 – 13625 485 125] --- We could test different amplification methods (keeping the reduction procedure) - for instance: a) instead of squaring the reduced digits, lift them to higher powers than 2 (d^3, d^4, d^5...) b) instead of powers, use multiplication (d becomes 2d, or 3d, or 4d, etc.) Is there a "more interesting" combination of reduction/amplification (in terms of results' variety)? Best, É.