How to solve route counting puzzles
Route counting: add the two ways in
Moving only right and down, how many routes cross the grid? Don't trace them — fill each square with the sum of the square above and the square to the left.
Updated
From the top-left square to the bottom-right, moving only right or down, past the occasional blocked square — how many distinct routes? Tracing routes one by one collapses beyond a handful. The counting rule does the whole grid mechanically, one addition per square.
The corner rule
The start square has exactly 1 route. Every other square's count is the sum of the square ABOVE it and the square to its LEFT — because when movement is only right and down, those are the only two ways in. Fill the grid square by square and the bottom-right corner IS the answer.
Blocked squares are zeros
A blocked square has 0 routes and passes 0 on to its neighbours. Write the zero and keep filling — do not reroute by eye. On blocked grids the signature wrong option is the clear-grid answer: what the count would have been if nothing were blocked.
Edges are the easy warm-up
The top row and left column have one way in, so they read 1, 1, 1, … — until a block appears, after which every later square on that edge is 0. Filling the edges first gives the interior a reliable frame.
The strategy
- Write 1 in the start square.
- Fill the top row and left column, honouring any blocks.
- Sweep the interior row by row: above + left, zeros on blocks.
- Read the bottom-right corner. That number is the answer — not a square count, not a step count.
Common traps
- Solving the grid as if it were clear.
- Counting squares, or the steps in one route, instead of whole routes.
- Missing that a block on an edge zeroes the rest of that edge.
- One addition slip mid-grid — every later square inherits it, so check the last row's arithmetic.
Speed · ~60s/item: Roughly 30–95 seconds. The fill is one addition per square — a 5×6 grid is under thirty additions, most of them trivial.
Try it yourself: 2 practice questions
Pick an answer, then reveal it. Nothing is recorded, there is no clock, and none of these questions is ever used in a test.
Moving only right or down, and never through a blocked square, how many different routes lead from the start to the finish?
● start · ◎ finish · ✕ blocked. Moves: right or down only.
Reveal the answer
The answer is E — 7.
Show why — the worked reasoning
Moving only right or down, and never through a blocked square, how many different routes lead from the start to the finish?
● start · ◎ finish · ✕ blocked. Moves: right or down only.
Reveal the answer
The answer is D — 4.
Show why — the worked reasoning
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Questions people ask
Will route counting questions be on the free test?
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