Tortoise & Hare · Aiming Lab

Playback Playback changes animation speed, not who wins.Ready to race

🐢

Tortoise

Precise, deliberate steps

Pending

🐇

Hare

Quick, uncertain steps

Pending

Tune the racers

Changes reset the race

Each step: aim toward 🏁, add any error, move, then aim again. Travel speed = step size × steps per second. Change one control at a time to see what it does.

World & finishing rules

Goal distance units

How far the goal’s center is from the start. Units are arbitrary distances, not meters or dollars.

Goal radius units

Size of the finish area. Touching its circular edge counts. A larger radius is easier to hit.

Time limit seconds

Stop observing at this simulated time. An unfinished racer stays unfinished; this does not automatically give the other racer a win.

Seed

A number that recreates the same random paths. “New seed” gives another race; it does not change either racer’s abilities.

Shorten steps near the goal

Cap step length at the remaining distance to the goal center. The step still takes the same interval; this can reduce overshooting.

Each vector takes one step interval to traverse. Crossing the goal disk counts as finishing. The plane has no walls.

Run more races

Run 500 trials with these settings, or compare the hare’s error and step frequency.

No trials yet. The same seed reproduces the same race.

What does this model assume?

At each step, the racer aims at the goal plus an independent normally distributed angular error. “Aiming error” is the standard deviation, in degrees. At 90°, roughly 32% of headings point backward. A constant bias adds systematic misalignment.

Steps start immediately and move at constant speed until the next re-aim. Finishing time includes the fractional step that first intersects the goal. No walls, obstacles, rests, or hill climbing are included. A time limit is a censored result, never an automatic loss to the other unfinished racer.

Far from the goal, average forward speed ≈ steps/second × step size × exp(−σ²/2) × cos(bias), with angles in radians. This is a local projection, not an exact prediction of finishing time. Large steps introduce sideways detours and make the last few units harder.

Batch trials use independent, reproducible random streams for the two racers. Medians describe finishers only; unfinished counts and a 95% Wilson interval for the hare’s win probability are shown separately.