The simplest computer that isn’t nothing: a handful of STATES and rules for jumping between them as it reads input, one symbol at a time. It remembers only which state it’s in — nothing else. Yet that’s enough to check a password format, run a turnstile, or here, decide if a binary number is divisible by three. Slide the input and watch the machine walk its states.
A deterministic finite automaton (DFA) is a finite set of states, a start state, accepting states, and a transition function δ(state, symbol) → state. It reads input left-to-right using only its current state as memory — recognising exactly the REGULAR languages. This one tracks a binary number mod 3: from state r on bit b, go to (2r+b) mod 3; accept iff it ends in state 0. A fail-loud self-check throws unless it accepts multiples of 3 (110=6) and rejects others (10=2). ◆ real automata theory, node-verified.
A DFA cannot count unboundedly or match nested brackets (that needs a stack / pushdown automaton, or a [[the-turing-machine|Turing machine]]); its power is exactly the regular languages. The mod-3 recogniser is exact.