Christmas Problem: Santa Claus Gifts

If on a map we mark each house in the world with a dot and join them by lines we would obtain a gigantic graph that would allow us to travel through all the houses in one route. In this graph, the houses would be the nodes of the graph. But this year’s problem is not about graphs. We are going to use a map of the Iberian Peninsula as an example:

Of course, the way to join these points is not unique: there are several (And many!) different ways to go through all these points.

After this initial approach our problem begins:

It’s Christmas Eve and Santa wants to stop by every house to give presents. Each year he takes a different route, but each year he starts with the house to the west and ends with the house to the north (which are always the same).

a) How many years will it take for Santa to repeat the route in your neighborhood? (10 houses).

b) How many years will it take for Santa to repeat the route in Spain? (26,018,179 houses).

c) How many years will it take for Santa to repeat the route in any country (n houses)?