Signal

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.

Every rule the rulebook permits 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 language is a free monoid generated by these three. The puzzle’s difficulty isn’t in any single rule’s complexity — it’s in how rules at later phases interact with rules at earlier ones.

Rule.comp (Rule.remove p) (Rule.remove p) = Rule.remove p

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 bits collected; when it fills, the rule is identified.

A practitioner’s sequence on this five-rule slice: empty mat first, then one of each. Two transmissions usually end the puzzle, sometimes one. Against the full 25-card deck the same sequence works because the deck’s response-fingerprints are even more spread out.

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 is a sum of those moves.

∀ (f : Token → Token) (m : Mat),
  (Rule.adjust f m).size = m.size

∀ (p : Token → Bool) (m : Mat),
  (Rule.remove p m).size ≤ m.size

∀ (extra : Mat) (m : Mat),
  (Rule.add extra m).size = m.size + extra.size

The empty mat partitions the catalogue by response size alone. Three rules return an empty mat; two return one token. The shape of that lone token splits the latter pair instantly.

Eight token kinds. Up to ten tokens on the mat. The number of distinct responses the Alien can produce is the stars-and-bars count of multisets of eight kinds with at most ten elements — multichoose 9 10 = 43 758. Each round carries about 15.4 bits of distinguishing power. The 25-card deck is 4.6 bits. Even the layered three-rule hypothesis space (25³ = 15 625) is under 14 bits. One transmission has the channel capacity to identify any of these.

Signalisn’t hard because information is scarce. It’s hard because the three rules compose, later rules re-shape what earlier rules did, and the social constraint of cooperative play limits communication to the mat itself. The channel is wide; the deduction is narrow.

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.

Knuth’s bound is the floor on rounds; the rulebook is the ceiling. The interesting question is whether the floor exceeds the ceiling for the catalogue at hand. For Signal, it does not — by a wide margin, and that margin is itself a kernel-checked theorem.

DeckSize ^ PhaseCount ≤
  Fintype.card (BoundedMat MatCap) ^ TransmissionBudget

The strategy is phase-aware. 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 job is to pin the new rule by sending inputs the previous phases’ rules already make predictable.

For the Experts: spend bits, not tokens

Three transmissions per phase fit comfortably in the budget: one to open the unknown, one to discriminate among the candidates, one to confirmthe prediction. Knuth’s minimax in plain English: send the transmission whose response you cannot already predict.

A clean run spends nine transmissions and banks twenty-one of thirty.

For the Alien: cooperate, but resist

You picked the card and you can’t describe the rules. Every response leaks information — and your timing leaks more than the rulebook says. Be readable enough to be solved within the budget, no more.

1. Pace uniformly
   Move tokens at the same speed whether the rule fires strongly or trivially. Slowing down when skipping a non-applicable rule telegraphs the negative case.
2. Pick rules that interact
   A phase-2 rule that commutes with phase 1 is easy to read in isolation. Pick something whose effect depends on phase 1's output.
3. Mask the size signal
   A pure removal or pure add changes the token count visibly. A remove-then-add of equal cardinality is size-neutral and hides the verb.
4. Goal on the manifold
   An unreachable goal wastes Expert transmissions on search, not deduction. Pick a goal a clean transmission can hit.
5. Don't volunteer
   The rulebook lets you correct recollection; it doesn't let you say hot or cold. Stay there.

What both sides share: every transmission is a question, and every response is information. Knuth’s minimax — pick the question whose answer you cannot already predict — is the same instruction in both seats. Your reward is shared; you both want the puzzle deduced, but only just.

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 the bit-budget that links them.

Signal’s puzzle isn’t mathematical scarcity; it’s structural commitment. You have the bits. Spend them well.

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