LCM Gears Game

How to Play LCM Gears

Set the displayed cycle periods so that the gears first align together at the target tick after zero. Each gear completes one independent cycle in its selected number of ticks. Choose a period from each menu, then move the inspection slider to examine the cycle positions. The marker returns to the top reference position at exact multiples of that gear’s period.

All gears begin aligned at tick 0, which does not count as the answer. The next shared alignment occurs at the least common multiple of their periods. Your task is to make that first positive shared alignment equal the target. Move to the target tick and select Check first alignment. The Target tick button offers a quick way to inspect that time.

The dials model independent repeating cycles; they are not a physical simulation of meshing gear teeth or torque. Their mathematical rule is exact: at tick t, a period p is aligned when t is divisible by p. The text beneath each dial states whether it is aligned, so the challenge does not depend on recognizing a color or estimating an angle.

Worked Example: First Alignment at Tick 12

A period of 4 aligns at ticks 4, 8, 12, 16 and so on. A period of 6 aligns at 6, 12, 18 and 24. Their first shared positive alignment is tick 12, so LCM(4, 6) = 12. Inspect tick 6 to see only the second cycle aligned, then tick 12 to see both markers return to their reference positions.

A shared multiple is not always the least one. With periods 3 and 6, both gears also align at tick 12, but they first align at tick 6. Therefore that setup is rejected when the required first alignment is 12. A useful check is LCM(a, b) = a × b ÷ GCD(a, b), which avoids counting a shared factor twice.

For three gears, combine the periods one at a time: LCM(4, 6, 9) = LCM(12, 9) = 36. A pair of periods with no shared factor greater than one has an LCM equal to its product. The listed first multiples help you compare choices, while the slider lets you inspect exact integer ticks without an animation or a speed requirement.

Multiple Solutions, Levels and Saved Progress

Easy offers two periods selected from smaller whole numbers. Normal increases the period choices and target range. Hard uses three cycles, which must all align for the first time at the required tick. Every generated puzzle has at least one verified setup. Other combinations are accepted when their exact least common multiple also equals the target.

Change menus by touch or keyboard. Focus the tick slider and use arrow keys for one-tick changes; Home and End reach zero and the target. The tick buttons provide the same inspection controls. Undo restores a previous setup or tick-button change and marks assistance. Slider inspection alone is not an incorrect answer. A hint supplies one valid setup but still leaves you to inspect and check it.

Daily and Practice sessions save separately by level, including chosen periods and the inspection tick. Wrong checks reduce the possible score by five points and each hint by fifteen, with a minimum score of ten for a completed round. There is no timer. Hints and Undo mark assistance, assisted rounds do not replace an unaided best, and the same completed puzzle is counted only once. When browser saving is blocked, all game controls still work for the current visit.

Frequently Asked Questions

Why does tick zero not count?

Every cycle starts aligned at zero. This task asks for the first positive shared alignment.

Why can a shared alignment be rejected?

A setup is rejected if the gears aligned together earlier. The target must equal the least common multiple, not just a common multiple.

Can more than one setup solve a puzzle?

Yes. Any listed combination whose exact least common multiple equals the target is accepted after inspecting that tick.