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Count the routes

Moving only right or down, how many routes run from A to B around the blocked squares — or through the ●? Count them exactly, square by square.

  • Spatial
  • Untimed
  • 14 levels

How it plays

A board with A in one corner and B in the other. Moving only right or down, count every route from A to B that never crosses a ■ — or, in the other question, every route that passes through ●.

Do not trace routes one by one. Every square is entered from above or from the left, so the number of routes into it is the sum of those two numbers; a ■ holds 0. Fill the board from A and read the count at B. For ●, count A to ● and ● to B and multiply.

Levels grow the board and add ■ up to about 8 × 8, where the counts reach the thousands; there the levels stop, and the counter says so.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This practises counting by building up from smaller counts — the add-the-neighbours table — on boards like these. It claims nothing wider.

Learn

How to get better at count the routes: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Read the question: routes that avoid every ■, or routes that pass through the ●. The second is two counts multiplied, not one.
  2. Put 1 on A. Fill the top row and the left column first: each square there has only one way in, so it holds 1 until a ■ — and 0 from the ■ onwards.
  3. Fill the rest row by row, left to right. Each square is the square above it plus the square to its left; a ■ holds 0. Say each sum under your breath, or point at it.
  4. Read the number on B. For ●, count A to ● with this table, then start again from ● with a 1 on it and count ● to B, and multiply the two.

Techniques and tricks

  • Add above and left

    Every route into a square arrives from above or from the left, never both, so the counts add. This replaces tracing routes one by one, which fails past a 3 × 3 board.

    e.g. Clear 3 × 3: rows 1 1 1 · 1 2 3 · 1 3 6, so 6 routes

  • Clear boards are choices

    With no ■, a route on an r × c board is r − 1 downs and c − 1 rights in some order. Count the ways to place the downs: C(r + c − 2, r − 1). Learn a few: 3 × 3 is 6, 4 × 4 is 20, 5 × 5 is 70.

    e.g. 3 × 5: C(6, 2) = 15 routes

  • Subtract the routes through a ■

    With only one ■, count the clear board, then take away the routes that would pass through the ■: routes to it × routes from it.

    e.g. 4 × 4 with ■ on row 2, column 2: 20 − 2 × 6 = 8 routes

  • Split at the ●

    Every route through ● is a route from A to ● followed by a route from ● to B. Multiply, because each first half goes with each second half.

    e.g. 3 routes to ●, 4 from ● to B: 3 × 4 = 12

  • Zero the shadow of a ■

    A ■ in the top row cuts off everything to its right in that row; a ■ in the left column cuts off everything below it. Write those 0s first so you never carry a 1 past a block.

  • Check from the other end

    Routes from A to B are routes from B to A walked backwards. Fill a second table from B, each square the one below it plus the one to its right; the number on A must match.

    e.g. Clear 3 × 3 from B: rows 6 3 1 · 3 2 1 · 1 1 1, and A holds 6

As the levels climb

  • Levels take turns: the board grows a column or a row, then the ■ thicken — from about a tenth of the squares to a quarter. More ■ make the count smaller but the table fussier, so fill the 0s first.
  • By 5 × 5 the counts reach the dozens and tracing is hopeless; the add-above-and-left table is the only reliable way. Keep each row’s numbers in a line you can say.
  • The ladder stops at 8 × 8, where clear counts run to thousands. There the slips are a single wrong sum, so check the last row twice: B is the square above it plus the square left of it.
  • The time grows by 3 seconds for every row and column, but the game is untimed unless you turn on Time pressure.

Traps to avoid

  • A ■ forgotten: the count with one block lifted. Before you add, cross the ■ squares off one by one so each one is a written 0.
  • The board treated as clear: the answer C(r + c − 2, r − 1) when there are ■ in the way. That number is only right with no blocks.
  • For ●, adding the two halves instead of multiplying them, or giving only one half. Every route to ● goes on by every route from ●.
  • For ●, giving all routes from A to B. Routes that miss the ● do not count.
  • Carrying a 1 past a block in the top row or left column. The squares beyond it are unreachable along that edge, so they hold 0; a stray 1 there inflates every count built on it.

How to practise

  • Tick only “Routes around ■” under Questions until the table is automatic, then add “Routes through ●”.
  • Set Answer by to “Typing the number”. The slips are among the options, and typing makes you finish the table instead of recognising a number.
  • When you miss, compare your rows with the table in the working, which lists every row of the board: the first square that differs is the slip.
  • Learn the clear counts once — 6, 10, 15, 20, 35, 70 — so you can see at a glance how much the ■ have taken away.

Levels

It is untimed unless you turn the clock on. There are 14 levels with the default settings, and the counter stops at the real top. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

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