Put two equal coins flat on a table, heads up. Hold one still and roll the other round its edge, without slipping, until it is back where it started. How many times did the moving coin turn? Most people say once, because the two edges are the same length. The answer is twice. The same extra turn is why the Earth in a Technic-style orrery spins 2⅔ times per trip round the Sun, not the 1⅔ the tooth counts seem to say.
The puzzle
Try it before reading on. Two coins of the same size are all you need. Put a pencil mark on the rolling coin, or watch the head on its face. Make one slow lap and count.
You will see the head turn upright twice. This is called the coin rotation paradox, and it is old enough that it once caught out the people who set a national exam: in 1982 a multiple-choice question on the US SAT used it, and three students showed that none of the listed answers was right, so the test was regraded.
The answer: one turn for rolling, one for going round
Two things happen at once:
- Rolling. The moving coin's edge rolls along the still coin's edge. Equal edges means one turn's worth of rolling.
- Going round. Even if it didn't roll at all, a coin carried once round a circle would end up turned once, the way you turn once when you walk round a table and keep facing its middle.
Add them: 1 + 1 = 2. For any sizes the rule is
turns per trip = (still gear's teeth ÷ moving gear's teeth) + 1
If you count while riding round with the moving coin, like an ant on the arm, the "going round" turn disappears and you only see the rolling part. Both counts are right; they are measured from different places.
Where it hides in the orrery
The Sun-Earth-Moon Orrery (level 4, about 35 minutes) is built around this puzzle. Its 40-tooth Sun gear is locked to the base, so it never turns. A 24-tooth gear carrying the Earth sits on the end of an arm, 4 holes from the Sun axle, because 24 + 40 teeth ÷ 16 = 4 holes. You push the arm round by hand.
The tooth counts suggest 40 ÷ 24 = 1⅔ Earth spins per orbit. The build file's own note says 2⅔, and the coin puzzle explains the gap: 1⅔ for rolling round the Sun, plus 1 for the trip. Its play notes suggest putting a sticker dot on one of the Earth's knobs and counting. That is the best way to settle an argument about it.
Try it in the lab below. Switch "Count from" to riding on the arm and watch the extra turn vanish.
Interactive: Count the planet’s turns. A planet gear goes round a sun gear on an arm, as in the Sun-Earth-Moon Orrery and Watt’s sun-and-planet beam engine. Hold the sun or stop the planet spinning, change the tooth counts, and predict how many turns the moving gear makes per trip round, from the table and riding on the arm.
The same trick, backwards: Watt's sun-and-planet gear
James Watt needed his steam engines to turn wheels, but someone else held the patent on the crank. His answer was the sun-and-planet gear: a gear fixed to the bottom of the connecting rod (the planet) rolls round a gear on the flywheel shaft (the sun).
The Sun-and-Planet Beam Engine (level 4, about 50 minutes) rebuilds it. Here the 24-tooth planet is pinned to the rod, so it can't spin; it just goes round. That flips the puzzle: now the still gear is the planet and the sun does the turning, 1 + 24 ÷ 8 = 4 times per pump. With two equal gears the flywheel turns twice for each up-and-down of the beam, which is the coin puzzle exactly.
Why engineers like it
That extra turn is free gear change. Planetary gears, where small gears orbit a central one, are packed into automatic car gearboxes, bike hubs and electric screwdrivers because they give a big change of speed in a small round space. The Sun & Planet Gears idea page has the short version, and why compound gears multiply reduction shows the other common way to get a big ratio.
Try it on the real build
- Before you build, write down a prediction for the orrery: how many Earth spins per orbit?
- Build it. The orrery's build page lists the Bevel-Gear Windmill and the Spinning Radar as builds to try first.
- Put a sticker on one knob of the Earth, line it up with something in the room, and push the arm round once, slowly.
- Now count again, but this time watch the sticker against the arm instead of the room. What changes?
Check yourself
Interactive: Check yourself. A few quick questions on the ideas in this guide, each with an explanation.
- A coin rolls without slipping all the way round an equal coin. How many times does it turn?
- In the orrery, a 24-tooth Earth gear rolls round a locked 40-tooth Sun. How many Earth spins per orbit?
- Why does the orrery’s Earth gear spin at all?
- In Watt’s sun-and-planet gear, the planet can’t spin. What turns instead?
