Grid Walking

Rectangular Grid Walk - Minimal Restrictions


Perry the Platypus is on a secret mission. He needs to move in the coordinate plane from (0,0)(0, \, 0) to (4,2)(4, \, 2) without passing through (3,1)(3, \, 1). If Perry only moves 11 unit at a time to the right or up, in how many ways could he complete his mission?

The Andromedan Trade Goods Association requires the following of its member worlds, whose locations can be plotted as a 100×100100 \times 100 grid.

  • Each time a world receives a trade good, it must send that trade good to one of the worlds immediately to the right of or above it.
  • Another trade good must be sent to the other world.

Unfortunately, the world Aberdeen, located 22 worlds to the right and 22 worlds above Bellerophon, has stopped complying with Andromedan regulation and does not pass on any trade goods, either new or received. All other worlds still comply with Andromedan regulation. If Bellerophon, located in the bottom-left corner, sends out a trade good to each of the two worlds next to it, how many trade goods will the world 66 to the right and 33 above the initial world receive?

An ant in the coordinate plane is located at (1,2)(-1, \, -2), and it can move repeatedly one unit to the right or up. If it wishes to travel to (3,3)(3, \, 3) without passing through the origin or (1,1)(1, \, -1), then how many possible paths could it take?

A particle is moving from the origin to (6,4)(6, \, 4). If the particle moves one unit at a time to the right or up and cannot pass through (2,1)(2, \, 1) or (4,3)(4, \, 3), how many possible paths could the particle take?

Micro Man is trapped in an incomplete sudoku grid! If he starts at the 99 and wants to travel to the 22 without hitting the 33, how many paths could he take, provided he wants to get there as quickly as possibly, while each step moving to a square sharing an edge with his current square?


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