Ways home
The lines are streets. You walk from Start to Home, and you may only go right or up. Tap any corner to see how many different routes lead to it. Guess the number before you tap.
What's going on
Think about the last step of a walk. You arrive at a corner either from the corner on its left or from the corner below it. There is no other way in. So the number of routes to a corner is the routes to the left corner plus the routes to the corner below.
That one rule fills in the whole grid. Along the bottom street and the left street every corner has just 1 route, because you can only walk straight. Every other number is the sum of two numbers you already know.
The numbers grow fast. A 2 by 2 grid has 6 routes home, a 4 by 4 grid has 70, and a 6 by 6 grid has 924. Nobody could list 924 routes without a mistake, but adding pairs of numbers gets there in a few minutes.
Turn your head and look along the diagonals: 1 1, then 1 2 1, then 1 3 3 1. This is a famous pattern called Pascal's triangle. See the Pascal's triangle demo.
Sheet: How many ways home?