Ant cell-state walk

Apply a turn–flip–move rule whose next action depends on a cell changed by earlier steps.

Adult planning suggestion: about 9–12 · Logic & rule checking · Shape & spatial thinking · Observation & recall

Players

Groups of 3

Time

12–18 minutes

Materials and quantities

One 7×7 white grid, an arrow marker, pencil/eraser and eight-line trace per trio.

Thirty pupils: ten trios; read, update and record roles rotate at stated steps.

Prepare

Start at D4 facing north on a completely white 7×7 grid, row 1 at the bottom. Use the source convention: on white turn left, on black turn right. In either case invert the colour of the cell being left, then move forward one square. A ninety-degree turn happens on the current square; a move happens afterward. Keep a visible arrow so direction is not held only in memory.

Play

Before each step, read the current square’s colour. Write the original colour, chosen turn, new direction and destination; then mark the departed square’s new colour. A black square becomes white, not a second black mark. One pupil reads the rule, one updates the board and one records the state; rotate roles after steps 3 and 6. Do not look at the destination’s colour until the next step.

Worked example

The first four destinations are C4 facing west, C3 facing south, D3 facing east and D4 facing north. D4 is now black because the first step changed it. Step 5 must therefore turn right, reaching E4 facing east. Repeating the first left turn would ignore the board’s stored state. The teacher table checks all eight steps, including both colour changes at D4.

Finish and check

Stop after eight steps for the supplied challenge and compare the recorded final cell, direction and black-cell set with the separate key. If any step would leave the paper in an extension, stop before that move and record the boundary; do not wrap to the other edge. A disagreement should be traced to the first inconsistent state, not corrected only by moving the final marker to the answer.

Adaptation and practical limits

For thirty pupils use ten trios. Adult planning suggestion is roughly 9–12; the mathematical source describes the automaton, not an evaluated primary lesson. Shorten to four steps before introducing the return visit if needed. Use shading and an explicit B/white note, with large squares. This finite eight-step paper trace does not demonstrate the long-run highway behaviour discussed in the reference.

Free alternatives: Use paper or clean reused cardboard and a pencil. Copy the supplied diagrams and tables by hand; no specialist kit or purchase is needed.

Printable materials

Preview at 100% scale, with browser headers and footers off. Cut with suitable help. Keep host keys separate. This is a manual tabletop kit, not an online opponent, automatic judge or solver.