Signal
The shape of a guess.
DVC Games · 2025 · cooperative deduction
The box ships 25 Alien cards. Each card holds three hidden rewrite rules and one printed goal signal. Phase 1 applies rule 1 only. Phase 2 layers rule 2 on top of rule 1. Phase 3 layers all three. In each phase the Experts have ten transmissions to make the Alien’s response match the printed goal.
Donald Knuth published a kindred finite-deduction problem in 1976. The counting comparison below is a necessary capacity check, not a rule for play, a sufficient condition for identification, or a strategy theorem.
Act 1— The mat
What the alphabet looks like
The game ships with a fabric mat and 40 signal tokens — rods, cubes, discs, and triangles, in black and white, eight kinds in total. Each Alien card holds three numbered rules and one printed goal signal. Phase 1 applies rule 1 only. Phase 2 layers rule 2 on top. Phase 3 layers rule 3 on top of those.
In the page’s finite model, each represented rule is a composition of three primitive verbs: adjust (rewrite each token via a function), remove (drop tokens matching a predicate), and add(append a fixed multiset). Adjustments preserve size; removals shrink it; adds grow it. The rule model’s rule language composes these operations. The puzzle’s difficulty is not captured by any single rule’s complexity — it’s in how rules at later phases interact with rules at earlier ones.
Eight tokens — click one
Audrey.Signal.Rule.remove_idemRule.comp (Rule.remove p) (Rule.remove p) = Rule.remove p
Removals are idempotent — applying the same removal twice is the same as once.
Act 2— Probe
The deduction, on a worked example
The published deck has 25 Alien cards. For teaching, we use a small five-rule sample below — Inverter (flip every colour), Sieve (drop white tokens), Pebbler (append a black disc), Twin (cubes → discs, rods → triangles), and Reaper (drop triangles, append a black cube). One is hidden. Transmit, read the response, narrow the partition. The bar above the controls tracks log₂ reduction in that five-rule candidate set. When one candidate remains, it is unique inside this demo only.
Probe the alien
send a probe
No transmissions yet. Pick a probe above.
A heuristic sequence for this five-rule slice is an empty mat, then one of each. Try it against the browser’s five hard-coded functions. No full-deck response table or injectivity proof is loaded here, so this example makes no claim about the published 25-card deck.
The “one of each” probe
Two colours, four shapes, eight combinations: black cube, black disc, black rod, black triangle, white cube, white disc, white rod, white triangle. Lay all eight on the mat and you have sent one of each. Audrey’s mom Jill says it about desserts; the probe says it about tokens.
In this five-function demo, the input exercises several modeled token cases. An Inverter flips all eight colours. A Sieve drops the four whites. A Twin remaps cubes to discs and rods to triangles. A Reaper drops the two triangles and appends. The resulting responses can be compared directly for these named functions; that observation does not quantify over the full rulebook.
As a teaching heuristic, pair an empty mat with one of each. In the demo, the empty mat exposes modeled additions while one of each exercises the displayed adjustments and removals. This is sample coverage, not a theorem that two transmissions identify every legal rule or composition.
Act 3— Size
Count the tokens, not the colours
One number on the mat is doing more work than any other: how many tokens come back. An adjustment leaves the count fixed. A removal can only shrink it. An addition can only grow it. A composition can mix those effects, so size alone does not recover the hidden rule.
Audrey.Signal.Rule.adjust_size∀ (f : Token → Token) (m : Mat), (Rule.adjust f m).size = m.size
Audrey.Signal.Rule.remove_size_le∀ (p : Token → Bool) (m : Mat), (Rule.remove p m).size ≤ m.size
Audrey.Signal.Rule.add_size∀ (extra : Mat) (m : Mat), (Rule.add extra m).size = m.size + extra.size
In the five-rule browser sample, the empty mat partitions candidates by response size: three return an empty mat and two return one token. The shape of that lone token separates the latter pair in this sample.
Of the five sample rules charted here, three return an empty mat. Pebbler and Reaper both add one token, distinguishable by shape.
Act 4— Numbers
What cardinality can — and cannot — say
Eight token kinds and the formal assumption MatCap = 10define a bounded-mat type with multichoose 9 10 = 43 758inhabitants. This per-response model parameter is distinct from the separate ten-transmission budget. Its log₂ cardinality is about15.4 bits. A set of 25 labels has log₂ cardinality about 4.6, and 25³ labels about 13.9. This inequality is a necessary counting bound only. It does not prove that every bounded mat is reachable as a response to one legal transmission, that distinct cards have distinct responses, or that any one-transmission identification strategy exists.
Grey bars are log₂ hypothesis counts. Teal bars are log₂ cardinalities of formal response or transcript types. The comparison is a necessary counting check only: it does not show that a legal probe reaches every response, separates every rule, or yields an identifying strategy.
Cardinality alone therefore does not explain practical difficulty. Later rules reshape earlier outputs, and cooperative play limits communication to the mat. Reachability, collisions between rule responses, and adaptive probe choice require evidence beyond the raw alphabet size.
Act 5— Budget
Ten tokens, three phases, thirty points
The rulebook caps round counts on both ends. The box ships ten transmission tokens per phase, three phases per Alien card. An Expert team has thirty transmissions total to solve a card; every unused token at the end is a point. Score is not win-or-lose — it is how cleanly you found the rule.
Each phase has ten transmission tokens. Grey marks the three-per-phase scenario drawn above; amber marks the remainder. This is a budgeting illustration, not evidence that every card or phase is solvable in three transmissions.
A transcript can identify a hypothesis only if its response alphabet is large enough to assign distinct transcripts. The displayed record checks the arithmetic inequality25³ ≤ 43 758¹⁰. That necessary room-in-the-alphabet condition is not sufficient: it constructs neither an injection from rules to legal transcripts nor an adaptive query strategy.
Audrey.Signal.Phase.layered_deck_cardinality_fits_full_trace_spaceDeckSize ^ PhaseCount ≤ Fintype.card (BoundedMat MatCap) ^ TransmissionBudget
Arithmetic only: 25³ ≤ 43 758¹⁰. The inequality compares cardinalities; it does not prove that legal probes realize separating transcripts or supply a solution strategy.
Act 6— Play
A heuristic playbook, on either side
The following rule of thumb is phase-aware. It is not derived from the cardinality inequality and is not certified to solve every card. In phase 1 you face one unknown; in phase 2 a known function and one unknown; in phase 3 a known composite and one unknown. Each phase’s heuristic goal is to probe the new rule using inputs whose earlier-stage behavior you can predict.
For the Experts: probe, compare, reassess
The diagram budgets three illustrative transmissions per phase: one to open the unknown, one to discriminate among current candidates, and one to check a prediction. A useful exploratory heuristic is to send an input whose modeled responses divide the remaining candidates; no worst-case bound is established here.
- OpenEmpty mat — can expose modeled additions
- DiscriminateOne of each — exercises modeled removals / rewrites
- ConfirmOne predicted input — compare before committing
- OpenAn input whose R₁-image you can predict
- DiscriminateVary inputs that R₁ collapses; compare candidate R₂ effects
- ConfirmPredict the layered response, transmit, check
- OpenAn input whose R₁∘R₂ image you can predict
- DiscriminateProbe the composite; compare candidate R₃ effects
- ConfirmCheck a predicted intermediate response
The pictured nine-transmission scenario would bank twenty-one of thirty tokens. It is an illustration, not a guaranteed score.
For the Alien: cooperate, but resist
You picked the card and you can’t describe the rules. A response can reduce the current candidate set, and timing can add an unintended side channel. A practical aim is to remain readable without turning timing into a separate code; the counting theorem does not guarantee a within-budget solution.
- Pace uniformlyMove tokens at the same speed whether the rule fires strongly or trivially. Slowing down when skipping a non-applicable rule telegraphs the negative case.
- Pick rules that interactA phase-2 rule that commutes with phase 1 is easy to read in isolation. Pick something whose effect depends on phase 1's output.
- Mask the size signalA pure removal or pure add changes the token count visibly. A remove-then-add of equal cardinality is size-neutral and hides the verb.
- Goal on the manifoldAn unreachable goal wastes Expert transmissions on search, not deduction. Pick a goal a clean transmission can hit.
- Don't volunteerThe rulebook lets you correct recollection; it doesn't let you say hot or cold. Stay there.
What both sides share is a practical need to compare predictions with observed responses. Choosing probes that split the current candidate set is a heuristic on this page, not a proved minimax policy for Signal. The cooperative reward remains shared.
Coda
The same machine, dressed twice
Knuth shipped his receipt for Mastermind half a century ago. DVC Games rebuilt the machine in 2025 with prettier pieces and a cooperative twist. Both run the same arithmetic: a hidden hypothesis, a public response, a finite budget, and a counting budget that can rule out alphabets that are too small without constructing a successful code.
Signal’s puzzle depends on structural reachability and separation, not just raw cardinality. The inequality clears one necessary counting obstruction; play must do the rest.
References.
Knuth, D. E. The Computer as Master Mind. Journal of Recreational Mathematics 9(1), 1–6 (1976–77).
Koyama, K. & Lai, T. W. An Optimal Mastermind Strategy. Journal of Recreational Mathematics 25(3), 251–256 (1993).
DVC Games. Signal — rulebook and Signal Support FAQ (2025). dvc.games/products/signal