Probability

# Rectangular Grid Walk - Minimal Restrictions

Perry the Platypus is on a secret mission. He needs to move in the coordinate plane from $(0, \, 0)$ to $(4, \, 2)$ without passing through $(3, \, 1)$. If Perry only moves $1$ 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 $100 \times 100$ 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 $2$ worlds to the right and $2$ 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 $6$ to the right and $3$ above the initial world receive?

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

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

Micro Man is trapped in an incomplete sudoku grid! If he starts at the $9$ and wants to travel to the $2$ without hitting the $3$, 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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