How to solve gear and pulley puzzles
Gears: parity gives direction, teeth give speed
Meshed gears alternate direction, so count the chain; belts copy direction unless crossed. Speed is teeth arithmetic. Two rules answer everything.
Updated
Mechanical-reasoning chains — gears driving gears, belts linking wheels, levers on pivots — reduce to two independent questions: which way does the last element turn, and how fast? Direction is a parity count. Speed is a ratio. Neither needs mechanical intuition once you know the two laws.
Direction: count the reversals
Every meshed contact reverses spin: gear 1 clockwise → gear 2 counter → gear 3 clockwise. So in a mesh chain, odd-numbered gears match the driver and even-numbered oppose it — position parity is the whole answer. Belted wheels turn the same way, unless the belt is crossed, which acts like one mesh: count each crossed belt as a reversal, each straight belt as none.
Speed: teeth in, turns out
Meshed gears pass teeth one-for-one, so turns scale inversely with tooth count: a 12-tooth driving a 36-tooth turns the big gear at one-third speed. Multi-stage chains multiply the ratios stage by stage. Bigger is slower — the intuition that big gears 'do more' is exactly backwards for speed.
Two facts answer most speed questions instantly
In a simple chain (one gear per shaft), turns × teeth is the SAME for every gear in the train. Two consequences fall out. First, the end-to-end ratio is just first teeth ÷ last teeth — everything in between cancels, so 'how many turns does the last gear make?' never needs the middle gears' sizes. Second, the gear with the FEWEST teeth is always the fastest and the one with the most is the slowest, wherever they sit. Questions about turn counts, the fastest or slowest gear, how many gears turn clockwise, or which way the largest gear spins are all read off these two facts plus the parity count.
The strategy
- Trace the drive path from the driver to the asked element, ignoring decorative gears off the path.
- Count reversals along the path (mesh = 1, crossed belt = 1, straight belt = 0): odd = opposite the driver, even = same.
- If speed is asked, chain the teeth ratios along the same path.
- Answer direction and speed separately — items love options that get one right and one wrong.
Common traps
- Off-by-one in the parity count — count contacts, not gears.
- A crossed belt read as straight (or vice versa) — look at the belt, not past it.
- 'Bigger gear = faster' — inverted; teeth ratio rules.
- Idler gears: they flip direction like any mesh but never change the overall speed ratio.
Speed · ~40s/item: Parity plus one ratio: 30–50 seconds for chains of any length. The only real risk is a rushed contact count.
Try it yourself: 4 practice questions
Pick an answer, then reveal it. Nothing is recorded, there is no clock, and none of these questions is ever used in a test.
Gear 1 turns clockwise (↻). How fast does the LAST gear rotate, as a multiple of gear 1's rate?
Numbers show each gear's tooth count. Gear 1 drives the chain.
Reveal the answer
The answer is D — ×9/20.
Show why — the worked reasoning
Gear 1 turns clockwise (↻). Which way does the last gear turn?
Numbers show each gear's tooth count. Gear 1 drives the chain.
Reveal the answer
The answer is A — ↻.
Show why — the worked reasoning
Gear 1 turns clockwise (↻). How fast does the LAST gear rotate, as a multiple of gear 1's rate?
Numbers show each gear's tooth count. Gear 1 drives the chain.
Reveal the answer
The answer is A — ×5/2.
Show why — the worked reasoning
Gear 1 turns clockwise (↻). How fast does the LAST gear rotate, as a multiple of gear 1's rate?
Numbers show each gear's tooth count. Gear 1 drives the chain.
Reveal the answer
The answer is D — ×7/8.
Show why — the worked reasoning
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Questions people ask
Will gears and pulleys questions be on the free test?
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