Rule garden / Experiment guide

One missing car, several disagreements

Imagine each lit cell is a car on a circular road. At each tick, every car with an empty cell immediately to its right moves one cell right. Everyone decides from the same old row, then moves simultaneously. This is rule 184. No car appears or disappears—but a missing car can change where several others end up.

Predict before counting

1. Open seed 7 on a 31-cell road. The original row has 15 lit cells. Flipping the center cell removes one car, leaving 14 in the second garden. Predict: must the two gardens always disagree at exactly one position?
2. Inspect generations 5, 10, and 15 in seed 7. The original occupancy stays 15, but the disagreement count reaches five. Switch between Original garden and Center cell flipped, pressing Grow each time, to see the two streams.
3. Open seed 12 at generation 5. This time the center flip adds a car: 12 become 13. There are already three disagreements where seed 7 still has one. The background affects the response.
4. Predict whether either pair can ever agree everywhere. Check generation 439, then read the conservation argument below: it says more than these 440 displayed rows can establish.

Scroll the table on narrow screens; keyboard users can focus it and use arrow keys. Original occupancy stays 15 for seed 7 and 12 for seed 12. The inspector’s lit-cell total always describes the original row, even when another view is selected.

Why the cars cannot disappear

Write xᵢ for a cell, either 0 or 1. Let Jᵢ = xᵢ(1 − xᵢ₊₁): it is 1 exactly when a car leaves cell i for its right neighbor. The update is x′ᵢ = xᵢ + Jᵢ₋₁ − Jᵢ, using ordinary integer arithmetic. A cell gains an arriving car and loses a departing one. The eight outputs, ordered from neighborhood 111 down to 000, are 10111000: binary 184.

Add the update over the whole ring. Every departure is someone else’s arrival, including at the seam, so the J terms cancel. The number of lit cells is conserved for every starting row, at every generation. This argument depends on the circular boundary; an open road could admit or lose cars.

Why five disagreements can mean one missing car

Let P count positions lit only in the perturbed garden, and M positions lit only in the original. Their occupancy difference is P − M; their disagreement count is P + M. Conservation keeps P − M equal to its initial value, either +1 or −1 after the center flip.

Therefore P + M is always odd and at least one. With seed 7 at generation 15, P = 2 and M = 3: five disagreements, but still just one fewer car. These are mismatched positions, not five lost cars. The two complete rows can never agree, because they contain different totals.

This is a nonlinear route to the odd-parity protection seen in rule 150. Conservation does not make rule 184 XOR-linear: on a three-cell ring, 100 advances to 010 and 010 to 001, but their combined row 110 advances to 101, not 011. Nor does it make the difference independent of the background, as the two seeds show.

Try another width and seed. Record occupancy and disagreements separately, and predict which quantities must stay fixed. A conserved total protects some information without preserving every position—or preventing a small change from spreading.

Reference: Wolfram Atlas: rule 184. The movement interpretation, conservation proof and signed-disagreement argument are derived here from its eight decisions.

Checked 2026-09-10: an independent simultaneous-car-movement calculation agrees with the garden on all 8,190 bit rows of widths 1–12; 17,600 original/perturbed row pairs through generation 439 at widths 2, 7, 31, 256 and 801 preserve occupancy and odd disagreement. All listed checkpoints pass. The algebra, rather than finite testing, supports the all-time claim.