THETIS: the adaptive online IQ test

Each row is a sequence of lattice panels, and each panel is made from the one before it by hidden laws: quarter turns, mirrors, wrapping slides and color swaps. Laws may alternate, grow step by step, combine, act separately on black and white tokens, follow a cue socket, or change from one row to the next.

The note above the puzzle says whether one law or several are at work and whether a law may change. Find the panel that replaces the question mark.

Each row is a ring of black and white cells: its right end touches its left end. Each row is made from the row above by hidden laws: slides, mirror images, inversion, or a local law in which every cell's next color depends on itself and its two neighbors. Laws may alternate, grow, change once or twice, depend on a feature of the row, or act separately on cells in even and odd places.

The note says whether one law or several are at work and whether a law may change. Find the next row.

There are four key cards: K1 one black circle, K2 two white triangles, K3 three gray squares, K4 four hatched stars. An oracle approves or rejects each card placed under a key card by a hidden sorting rule: by shape, fill or count, possibly with a fixed offset, always the same key, or an if–then rule. The rule may change, return, or cycle through shape, fill and count.

The ledger lists earlier cards in order, where each was placed and the verdict. Say under which key card the oracle will approve the next card.

Machines combine two inputs into one output, and at any moment every machine obeys the same law of combination (for example keep everything, keep only what is common, or let what is common cancel). Machine A works on lattices, B on bead strings and C on tile rows, and a machine may record presence with the opposite color.

The law may change between runs or repeat in a cycle. Runs are shown in the order they happened. Give the output of the final run.

A board of lamps changes tick by tick. Each lamp's next state is a fixed hidden function of at most two lamps' current states (copy, opposite, both, either, exactly one, and so on; a lamp may watch itself), the same in every run. A lever can hold one lamp on or off for a whole run. In some puzzles one lamp is hidden from view, but it still takes part.

The simplest wiring is the one with the fewest input wires. The question under each puzzle says what to find: a later board, a starting board, a lever, or what would have happened.

Marbles roll through a machine one at a time, from an entry at the top into a numbered bucket. Each gate sends a marble left or right, then either flips after every marble, flips only after every second marble that passes it (a slow gate), or never moves (a stuck gate). Gates are drawn without their directions; the record shows where earlier marbles entered and where they came out.

The note says which kinds of gate can occur. Assume as few exceptions as the record allows: a stuck gate counts one, a slow gate two. When the note says a gate broke, take it as given that one gate which flipped after every marble became stuck for good during the record, and keep the other exceptions as few as possible.

Each token moves around its own ring of eight positions (one on each spoke) by a steady number of steps per tick, clockwise or counterclockwise. When a token arrives at the same spoke as another token, the two meet. If the note says a meeting rule is at work, one fixed hidden rule acts at every meeting: both tokens turn back, or the two exchange speeds, or both rest for one tick. Without that note, tokens never affect each other.

A dashed ring holds an invisible token that moves and meets like the others. Snapshots show the rings at ticks 0, 1, 2 and so on; choose the picture of the asked tick.