Phase 1: Try to count paths on grids that keep growing.

Count the paths you can

A robot sits in the top-left corner of a grid. It can only move RIGHT or DOWN. How many unique paths lead to the bottom-right? On a tiny grid you can count on one hand. Keep going, and the count detonates.

FIG. 1 — A 2×2 GRID. 2 UNIQUE PATHS.
2 × 2
A 2×2 grid — count on one hand.
2 paths