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Puzzle Foundry / PF-02 / The Parity Maze

The Parity Maze

A living six-room probability maze. Set the walkers loose, watch the two room-families pulse in perfect lockstep, then change one tiny rule and watch the same hidden balance suddenly become an attractor.
Stationarity · periodicity · convergence
What to watch

A crowd can move forever while its pattern holds.

Think of a hundred silver walkers moving at once. The room percentages are the pattern. Your first job is simply to watch that pattern breathe; the mathematics can open afterward.

Tiny rehearsal / two rooms

A distribution can stay the same while everybody moves.

Suppose rooms X and Y have one door between them. Under the ordinary rule, every walker must cross that door each step.

50%Room X
↔
50%Room Y
Stationary distribution: a set of room percentages that is unchanged after one application of the movement rule. Here, 50/50 becomes 50/50 after the walkers swap rooms.
Start at the balance pattern50/50 → 50/50. The distribution is stationary.
Start somewhere else100/0 → 0/100 → 100/0. Same rule, but this start does not settle.
Quick check: why can 50/50 be stationary if every walker moves?
Pick one. This does not affect your score.
Your six-room maze / first two steps

"Mass" is just the fraction of walkers in each room.

In the real maze below, all walkers start in A. A has only one exit, so the first move is forced. B has three exits, so the second move splits the group three ways.

Step 0

100% begin in A.

A 100%B 0%
Step 1

A can only lead to B.

A 0%B 100%
Step 2

B has exits to A, C, D.

A 33%C 33%D 33%
Ordinary walk: at every step, each walker must leave the current room and choose uniformly among that room's exits. 50/50 stay rule: flip a fair coin first; half the time stay, half the time make the ordinary move.
Now predict

Four guesses before you operate the real maze.

You have enough vocabulary now. These are predictions, not a test. The point is to compare what you expected with what the maze actually does.

1. Could there be a special split of walkers among A-F that looks exactly the same after one ordinary move?
Technical name: does a stationary distribution exist?
2. If 100% of the walkers start in A, will their room percentages eventually settle to that special balance pattern?
Technical name: does the distribution converge from the point mass at A?
3. If each walker gets a 50% chance to stay before moving, will the special balance pattern itself change?
This is the lazy rule used here: P_lazy = 1/2 I + 1/2 P.
4. With that 50/50 stay rule, will a group starting 100% in A eventually settle to the balance pattern?
Same starting condition; only the transition rule changes.
Choose one answer for each question.
Current rule
ORDINARY - must move - no self-loop
Step
0
Instrument live: A begins with 100% of the walkers. Press Step once, then run the maze and watch which family lights up on alternating beats.
Six-room parity maze Rooms A through F connected by edges A-B, B-C, B-D, D-E, D-F. Probability mass begins at A.
Temporal behavior changed
The maze settles.
Same stationary balance. The strict left/right march is gone.
Preview

See the map first.

A is marked START and currently holds 100% of the walkers. Lock your predictions above; then your first action will simply be one Step.
Math lens

Every symbol points to something you can see.

You do not need to memorize these equations. Use them as captions for the experiment: the notation is another view of the moving probability mass.

One step
pn+1 = pnP
pn is the six room percentages now. P is the movement rule. Pressing Step applies that rule once.
Balance pattern
πi = deg(i) / 10
π = (1,3,1,3,1,1) / 10
There are five edges, so the six degrees sum to 10. Rooms with three doors get three times the stationary weight of rooms with one door.
Why ordinary never settles from A
p2k(L)=1  ·  p2k+1(R)=1
Even steps live entirely in L; odd steps live entirely in R. Yet the stationary law itself has π(L)=π(R)=1/2.
Change one rule
Plazy = ½I + ½P
πPlazy = π
Half stay; half make the ordinary move. The target balance survives, but the strict L/R alternation is broken.
Distance diagnostic
D(pn,π) = ½ Σ |pn(i) − π(i)|
Distance diagnostic: total-variation distance. 0 means the current percentages exactly match the stationary balance.
The whole puzzle in two equations
πP = π  ≠  pn → π
A distribution can survive one transition unchanged without attracting this particular start. Stationarity and convergence are different questions.
SYMBOL KEY   pn = distribution after n steps · π = stationary distribution · deg(i) = number of doors at room i · I = “stay where you are” · P = ordinary movement rule
Concept check / no free-text box

Did the experiment change how you think?

Choose an answer; the instrument tells you why it is right or wrong. You can retry. Nothing is uploaded or silently stored.

Understanding matrix0 / 4

1. What does "stationary distribution" mean here?

Choose one.

2. What does "converges from A" mean?

Choose one.

3. Why does the ordinary walk from A fail to settle?

Choose one.

4. What does the 50/50 stay rule change?

Choose one.
The goal is 4 / 4 concepts, not a high score. Correct answers light the matrix above; wrong answers explain what to reconsider.
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