A gear train puzzle asks you to assemble several ideal external spur gears into a continuous contact chain so the driver's clockwise rotation reaches the target gear, with five deterministic levels worth exactly 200 points each for a 1,000-point total. Every contact is an external mesh between two pitch circles whose centers sit the sum of their radii apart within the game's small declared tolerance, and every external mesh reverses rotation. That single rule — odd chains flip the target, even chains match the driver — is the entire physics behind the puzzle, and it is also why a stuck puzzle can almost always be solved by counting gears or by tracing the contact graph. The Gear Rotation Puzzle expresses this rule with five fixed gearboxes, each offering three or four candidate gears that differ in tooth count and visible size, and a small empty rack that leaves no hidden math for the player to do. What changes between levels is the geometry of the board and the available tooth counts, not the rule.
How a Gear Train Puzzle Models Real Gears
A gear train puzzle models an ideal assembly of external spur gears. The board shows a fixed clockwise driver on the left, a target gear on the right, and a chain of empty sockets between them. Each candidate gear has its own tooth count and a matching pitch radius. Two gears are said to mesh when the distance between their centers equals the sum of their pitch radii within a small declared tolerance. The source fixture sets every level, so the same physical board appears on every run.
The single rule that drives the entire puzzle is direction reversal at every external contact. If one gear turns clockwise, the gear that touches it on the outside must turn counterclockwise, and the next gear back returns to clockwise. The signed speed at each neighbor follows a fixed rule: omega_next equals minus omega_current times teeth_current divided by teeth_next. That ratio is what the board reports beside the target once a connected path is found, and it is the same rule used in MIT OpenCourseWare's gears and linkages lecture for simple gear stages.
Read the Board Before Placing Any Gear
Before tapping a socket, read the visible geometry. Each candidate gear's pitch circle is drawn at the same module, so a gear with more teeth also has a larger radius. Two gears mesh only when their drawn circles actually touch — a small visible gap is enough to break the contact graph. If the leftmost socket sits very close to the driver but the next socket sits further away, the candidates in those two sockets must be sized so the distance between their centers equals the sum of their pitch radii.
Counting candidate teeth is the next step. The driver turns at a fixed angular speed and the target's expected speed depends only on the driver and target tooth counts once the idlers have been accounted for. Any gear placed between them contributes one reversal but cancels from the end-to-end magnitude, so the chain's overall ratio is fixed before placement. Use that fact to choose which gears go next to the driver and which go next to the target — and use the visible gaps to verify that the chosen pair actually meets the contact rule.
Solve a Gear Train Puzzle in Six Steps
- Open the Gear Rotation Puzzle at its fixed level-one board and identify the clockwise driver on the left and the target on the right.
- Pick the candidate gear from the rack whose tooth count and pitch radius match the gap between the driver and the leftmost empty socket, and place it into that socket.
- Repeat with the next socket, choosing each candidate so its pitch circle reaches the previous gear's edge. Fill every socket before testing.
- Press the test key or tap the test button. A correct, conflict-free path makes the target visibly turn, and the board reports clockwise or counterclockwise plus the ideal target-to-driver speed magnitude.
- On a failed test, check the board message for one of three outcomes: the target is disconnected, a placed gear sits isolated, or two paths assign the same gear a contradictory direction. One distinct failed layout is recoverable.
- Swap or remove the gear that breaks the contact graph, retest, then advance. Solving all five fixed gearboxes awards exactly 200 points each, totaling exactly 1,000 points.
Speed, Idlers, and the Cancellation Rule
The end-to-end speed of a gear train depends on driver and target tooth counts only when intermediate gears cancel. The table below summarizes the four cases a player is likely to encounter while reading a gear train puzzle.
| Chain shape | Direction at target | Effect on speed magnitude |
|---|---|---|
| Two external gears | Reverses once | Driver teeth divided by driven teeth |
| Three external gears | Same as driver | Driver teeth divided by last-gear teeth |
| Add an idler between driver and target | Adds one extra reversal | Unchanged — idler cancels |
| Odd loop of external gears | Parity conflict, unsolvable | Conflict — layout cannot score |
A worked example shows the cancellation in action. With a 24-tooth driver turning at signed speed +1, a 12-tooth first neighbor reaches -1 × 24 / 12 = -2, and a 24-tooth second neighbor at the end of the chain reaches -(-2) × 12 / 24 = +1. The 12-tooth intermediate gear appears once in the numerator and once in the denominator along the chain, so it cancels from the final magnitude.
An idler is still useful: it bridges a physical gap and contributes one reversal, so a chain that would otherwise be too short can still flip or match the driver. The published treatment of gear stages describes the same cancellation that this puzzle applies.
Parity, Loops, and the 200-Point Ladder
The solver does not compare a placement against a hidden answer key. Instead, it propagates signed speed through the contact graph with a breadth-first search, beginning at the driver with +1, visiting every geometric neighbor, and assigning each neighbor its expected opposite direction and tooth-count-adjusted speed. If a later path reaches an already-assigned gear with a contradictory direction or speed, the placement contains a parity conflict and the target stays stopped.
An odd loop is the classic trap: returning to the starting side after an odd number of reversals asks one gear to rotate both ways. An even loop can remain consistent because every contact adds two reversals around the cycle. Players can use this observation to prune wrong layouts without testing — if a candidate chain forms an odd triangle among three already-placed gears, swap one of the three before pressing test.
Scoring is straightforward and deterministic. Each correct layout awards exactly 200 points. There is no time bonus, no random level, no adaptive difficulty, no hidden multiplier, and no penalty for exploring placements. Five correct layouts therefore produce exactly 1,000 points, and submitting the identical failed assignment twice does not consume a second chance — a second distinct failure closes the run.
Keyboard Controls and Reduced-Motion Access
The full construction loop runs without a pointer. Number keys 1 through 4 hold the available candidate gears; Q, W, E, and T address the four sockets from left to right. Pressing a socket key while a candidate is held places the gear; tapping an occupied socket while no candidate is held clears it. Enter tests a fully filled train and R restarts from level one with an empty board, zero score, no failed signatures, and no held candidate. Every on-screen target is at least 44 pixels with visible focus and held states.
Reduced-motion users receive the same direction and ratio state without depending on animation. The final completed level remains visible and rotating while every interaction is frozen, so the 1,000-point outcome matches what the board actually shows rather than replacing it with an empty result screen. The game runs locally in the browser, with no account, download, upload, API, or remote service involved.
What This Puzzle Is Not
This is an educational placement puzzle about direction and ideal ratios, not engineering software. The Gear Rotation Puzzle deliberately omits torque, power loss, tooth stress, backlash, profile interference, shaft alignment, lubrication, noise, wear, materials, safety factors, tolerances, and dynamic loads. The scope is shown beside the board and in the methodology; the puzzle must not be used to size a real transmission. The five fixtures are deterministic original layouts, and their candidate radii, tooth counts, socket coordinates, driver dimensions, and target dimensions do not change between runs.
For a deeper look, see Gomoku Rules: How to Play Five in a Row on a 15×15 Board.
For a deeper look, see Heyawake Rules: How Room Shading Logic Puzzles Work.