Finite-state door
Follow every symbol through an explicit transition table and test the ending state.
Adult planning suggestion: about6–9 · Logic & rule checking
- Players
- 2–3 per group
- Time
- 12–18 minutes
- Materials
- State cards E/O, eight A/B cards, arrow table and record sheet.
Prepare
The door accepts exactly the messages with an even number of A letters. Start in E, also the accepting state. From E, A leads to O and B stays at E. From O, A returns to E and B stays at O. Draw all four arrows before testing any message.
Play
Lay a message left to right. Start a counter on E. Read exactly one symbol, take its specified arrow, and record the new state. Never skip B: it has a self-loop that still consumes the letter. Acceptance is checked only after all symbols are read.
Worked example
ABAA gives E→O→O→E→O, so it is rejected. ABAB gives E→O→O→E→E, so it is accepted. The empty message is accepted because the starting E is already accepting and no symbol changes it.
Finish and check
Swap four test messages with a partner and reveal both the state trace and the A count. Disagreement is settled by replaying the table, not by guessing the designer’s intention. Finish when each learner has traced an accepted and a rejected message.
Adaptation and limits
For thirty learners use ten trios with pointer, reader and checker roles. Use printed letters as well as state names and allow adult reading. This tiny machine is a teaching convention; it is not a physical access lock or a secure password system.
Purpose, observation and provenance
Within this activity: Follow every symbol through an explicit transition table and test the ending state.
Ask whether a temporary visit to E guarantees acceptance if more symbols remain.
Our rules, examples and printable diagrams are separately prepared. The earliest inventor of the broader mechanism is not established here.
Computer Science Field Guide — Finite state automata — Technical state-transition background; our parity-door table and task are separately written.
Age bands are planning suggestions. Goals describe observable actions, not measured long-term gains or institutional endorsement.
Free alternative: Copy the supplied grids, labels and cards onto reused paper. No specialist product is needed. Adults may prepare pieces or act as a scribe.
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.
Player materials · Finite-state door
Transitions: E+A→O; E+B→E; O+A→E; O+B→O. Start E; accept E.
Test: ABAA / ABAB / empty / AAAABB; write every state.
| State | A | B |
|---|---|---|
| E | O | E |
| O | E | O |
Player record · keep the host key separate
| Input | State before | State after |
|---|---|---|
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
Host key · separate from players
Reject ABAA; accept ABAB, empty and AAAABB. Count A modulo 2.
Run the round
The door accepts exactly the messages with an even number of A letters. Start in E, also the accepting state. From E, A leads to O and B stays at E. From O, A returns to E and B stays at O. Draw all four arrows before testing any message.
Lay a message left to right. Start a counter on E. Read exactly one symbol, take its specified arrow, and record the new state. Never skip B: it has a self-loop that still consumes the letter. Acceptance is checked only after all symbols are read.
ABAA gives E→O→O→E→O, so it is rejected. ABAB gives E→O→O→E→E, so it is accepted. The empty message is accepted because the starting E is already accepting and no symbol changes it.
Swap four test messages with a partner and reveal both the state trace and the A count. Disagreement is settled by replaying the table, not by guessing the designer’s intention. Finish when each learner has traced an accepted and a rejected message.
For thirty learners use ten trios with pointer, reader and checker roles. Use printed letters as well as state names and allow adult reading. This tiny machine is a teaching convention; it is not a physical access lock or a secure password system.