IBM Ponder This #213 · January 2016

As part of IBM's efforts to make a better and greener planet, this month's challenge is to design an efficient gear system for bicycles. The front chainrings (S_1) and back cogset (S_2) are represented by two sets of numbers S_1 and S_2 where each gear g is the product of two numbers, one from each set: s_1*s_2. When designing the gears, we need sets of numbers that can produce every gear value between 1 and N up to +/- 1, i.e., for every number m between 1 and N there exists s_1 in S_1 and s_2 in S_2 such that abs(m-s_1*s_2)<=1.

So, for example, using sets of size 3, we can get to N=19 by S_1={1,3,6} and S_2={2,3,5}.

Your challenge is to find the values of S_1 and S_2 of size 6 that solve the problem for N=56 (again, using +/- 1); Earn an asterisk for a significant increase of N.

Solution

Best opened after a real attempt

To be added.