The Billionaire's Gates

Lamentably for the billionaire, the answer is no. There is no route which passes through each gate once and only once. This puzzle is based on the classic "Seven Bridges of Konigsberg" geometry problem. The original puzzle asked if it was possible to traverse all the bridges of Konigsberg in a single trip without repeating any.

The first solution to the problem is credited to Leonard Euler (Pronounced with an "Oi" as in "oil"), while he was beginning to study the topology of networks.

His logic, translated to the problem of the Billionaire's Gates was as follows:

All the areas are contected to each other by a number of paths. After entering an area, one must leave by a different path. Any area with an odd number of exits must be passed in or out of an odd number of times. Except for where you start and finish, each area must be exited as many times as it is entered. Therefore, if a topology (or map) is to satisfy the criteria of having each of its paths crossed once and only once, there may be no more than two areas with an odd number of exits: where you start and where you finish.

The toolshed has four regions with an odd number of exits (remember that "outside" counts as a region), so the client's request is impossible.