The rules are as following:
1.Each gear has a width of 1
2.There are two shafts that have a fixed distance d between them
3.To change speeds, you can only slide one of the shafts to make gears of different sizes mesh together
4.Two gears can only slide past each other if the sum of their radii $\leq$ d
5.For each different speed, the gear ratio of the gears meshing together must be different
6.You always want the narrowest gear configuration possible to save space
What is the least possible width of a gear configuration that allows n different speeds?
