In short. Racetrack is a two-player pencil-and-paper game played on graph paper. Each player draws a circuit, then moves a car one grid point at a time. The catch is that your move is a vector: whatever direction and speed you had last turn carries over, and you may only adjust it by one square horizontally and one vertically. Leave the track and you lose a turn. First car across the finish line wins.
The game was invented in the 1960s and reached a wide audience through Martin Gardner's Mathematical Games column in Scientific American, January 1973. It has been rediscovered many times since, under names like vector racing or, in French, jeu de course papier.
What you need
Graph paper, one pen per player, and two minutes to draw a circuit. Draw a closed loop — any shape, as long as it has an inside and an outside — and mark a start line across it. Wide corners make an easy track; tight hairpins make a cruel one.
Each car starts on a grid point on the start line, at a standstill.
How a move works
This is the whole game, and it takes one paragraph.
Your car has a velocity: how many squares it travelled horizontally and vertically on the previous turn. On your turn, you may change each of those two numbers by at most one. Then the car moves by the new velocity, and you mark the new point.
From a standstill, your first move can only be one square in any direction — including diagonally. From there speed builds up one square per turn, and that is where the trouble starts: a car doing six squares a turn needs six turns to stop, and cannot turn sharply at all.
Leaving the track
If the straight line between your last point and your new one crosses the edge of the circuit, you have crashed. The usual penalty is the gentlest one: your car returns to the last point that was on the track, its velocity resets to zero, and you miss a turn. Harsher house rules exist — some groups simply eliminate you.
Two cars may not occupy the same grid point on the same turn. Beyond that, they do not collide: this is a race against the geometry, not a contact sport.
The one thing that makes it interesting
Because you can only change speed by one square per turn, braking has to be planned several turns before the corner. A beginner accelerates down the straight and arrives at the hairpin doing seven squares a turn, with no legal move that keeps them on the track. The skill of the game is not steering — it is knowing, four turns early, what speed you will need.
That is also why the game rewards taking the racing line. Cutting a corner tightly costs you speed on the way in; going wide keeps your velocity but travels further. It is the real trade-off of motor racing, produced by a rule you can write on one line.
Before each move, ask what velocity you want to be carrying when you reach the next corner — not this turn. Then count backwards. If you need to be down to two squares a turn in three moves, you can afford at most one more turn of acceleration.
How Dotrace adapts it
Dotrace keeps the constraint and drops the grid. Instead of marking discrete points on graph paper, you place control dots anywhere on the circuit and the ball follows a smooth curve through them — a Catmull-Rom spline — still respecting inertia: the faster it travels, the wider it has to turn.
Each dot you place counts as one move, and every circuit has a par, the number of moves a clean line should take. Beating par is the equivalent of the paper game's "fewest turns" goal.
The trade-off is deliberate: the smooth version is easier to pick up and reads better on a phone, at the cost of the crisp integer arithmetic that makes the paper original a favourite of maths teachers. If you want the exact original, graph paper is still the best implementation.
Frequently asked questions
Who invented the Racetrack game?
Its exact origin is unclear — it circulated among students and engineers through the 1960s. It became widely known when Martin Gardner described it in his Mathematical Games column in Scientific American in January 1973.
What are the rules in one sentence?
Move your car by its previous velocity, adjusted by at most one square horizontally and one vertically, without crossing the edge of the track — first to the finish line wins.
Can you play Racetrack alone?
Yes. Solo, the game becomes a puzzle: complete the circuit in as few turns as possible. That is the form Dotrace uses, with a par for each circuit.
What happens if you crash?
In the most common version, the car returns to the last point on the track, its velocity resets to zero, and the player misses a turn. Some groups eliminate the player instead — agree before you start.
How is Dotrace different from the paper game?
Dotrace replaces the integer grid with a smooth spline: you place dots freely and the ball curves through them. Inertia still governs how sharply it can turn, but the movement is continuous rather than square by square.
Play Dotrace
Handcrafted circuits, a par on every one, and no account to create. Free in the browser, or in the app.