examples.md

Worked example (K=3, T=6)

Prompt

3 cells sit in a ring (c1..c3, and c3 is adjacent to c1). Initial values: c1=6, c2=6, c3=0.

Each step, every cell SIMULTANEOUSLY becomes the sum modulo 10 of its two ring neighbours (left and right). Apply this for 6 steps. Only AFTER your thinking, you will be asked for one cell's final value -- answer with ONLY that value as \boxed{d} (a single digit).

Reason inside <think> </think> -- after each step write all 3 cell values in order c1..c3.

Hidden CA evolution (each row c_t carried by ONE soft token z_t)

c_0 (initial): [6, 6, 0] c_1 (after step 1): [6, 6, 2] c_2 (after step 2): [8, 8, 2] c_3 (after step 3): [0, 0, 6] c_4 (after step 4): [6, 6, 0] c_5 (after step 5): [6, 6, 2] c_6 (after step 6): [8, 8, 2]

Query

Final value of cell c2: -> 8