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, and the black and the white tokens may each follow a schedule of their own (for example, one color's law grows step by step while the other color's law follows a cue socket). One law may join a turn, mirror or slide with a color swap. A cue socket acts by being filled or empty, whatever the color of its token.

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, each set of places with a schedule of its own (for example, one set slides further at every step while the other keeps one law).

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. A rule that has changed, or changed back, then stays; a rule that cycles moves on after the same number of approvals each time.

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: a fixed rule that gives each cell of the output from the same cell of the two inputs. Any such rule can occur, for example keep everything, keep only what is common, let what is common cancel, keep what the first input has and the second lacks, or mark only the cells both inputs leave blank. Machine A works on lattices, B on bead strings and C on tile rows. A machine may record presence with the opposite color; it then does so in both inputs and in its output alike, in every run, and each such machine counts as one exception, added to the number of laws.

The law may change between runs or repeat in a cycle of any length. Once a cycle has been seen twice in full (each run's law read through its machine's colors), it counts: it goes on, and the changes within it are not counted. Until then, a law that changed and changed back stays: after the laws X, Y, X the law stays X, while after X, Y, X, Y the cycle counts and the next law is X. 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. A slow gate may already be partway through its count when the record starts.

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; a token that does not move arrives nowhere, so tokens resting on one spoke do not meet again. 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 and directions, 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.