Gear Train Puzzles: Which Way Does the Last Gear Turn?

In a gear train puzzle, two gears that mesh always turn in opposite directions, so the direction flips at every contact as you move along the train. Count the contacts from the gear you know: an even count means the same direction, an odd count means the opposite one, and a closed ring of an odd number of gears cannot turn at all.

That single rule is the heart of every gear direction puzzle. This guide covers how to trace a train, how to spot a jam before you start, how belts change the rule, how to work out speeds from tooth counts, a full worked example and a practice puzzle with its answer.

What is a gear train direction puzzle?

A gear train puzzle presents a set of gears touching each other, often with a handle or crank on one of them. You are told which way one gear turns, usually clockwise or anticlockwise. The question asks which way another gear turns, whether a weight on a drum goes up or down, or whether the whole set can move at all.

The gears are drawn flat, each on its own fixed axle. Touching gears mesh: the teeth of one sit in the gaps of the other, so neither can move without the other.

Some puzzles add belts, which join two wheels that do not touch, and some add tooth counts so that you can ask how fast a gear spins as well as which way.

Why do meshing gears turn in opposite directions?

Picture two gears side by side. Where they touch, the teeth of both gears must move together in the same direction, say downward. For the left gear, teeth moving down on its right-hand side means it turns clockwise. For the right gear, teeth moving down on its left-hand side means it turns anticlockwise.

So every contact between two gears is a flip. Gear one clockwise means gear two anticlockwise, gear three clockwise again, and so on down the line. A gear placed in the middle of a train only to pass the motion along is often called an idler, and its only effect on direction is that one extra flip.

How do you trace the direction along a train?

Treat the drawing as a network: each gear is a dot, and each pair of touching gears is a line between two dots. Then follow these steps.

1. Label every gear. Give each gear a letter so you can talk about it.
2. List the contacts. Write each pair of touching gears, such as AB or DE. Check the picture carefully, because two gears that look close may not actually touch.
3. Mark the known gear. Write CW for clockwise or ACW for anticlockwise beside the gear the puzzle tells you about.
4. Spread outward. Every gear touching a marked gear gets the opposite mark. Keep going until every connected gear has a mark.
5. Check every contact. Look at each pair on your list. If any pair has the same mark on both gears, the train is jammed.
6. Read off the answer. The mark on the gear you were asked about is the answer, as long as step five turned up no jam.

There is a shortcut for step four. Count the fewest contacts between the known gear and the gear you want. If that count is even, both gears turn the same way. If it is odd, they turn opposite ways.

Why does a ring of an odd number of gears jam?

Put three gears in a triangle so that each one touches the other two. Say the first turns clockwise. The second must turn anticlockwise, and the third must turn opposite to both of them. No direction can be opposite to clockwise and to anticlockwise at once, so the three gears lock solid.

The same thing happens in any closed ring with an odd number of gears. Going round the ring, the marks alternate, and with an odd number of gears the last mark lands back on the first gear with the wrong sign. A ring with an even number of gears comes back in step and turns freely.

What does a working gear train have in common with a two-colour network?

The marks CW and ACW are a two-colouring of the network: every line joins a dot of one colour to a dot of the other. Mathematicians call a network that can be split into two sides this way bipartite. A gear train turns freely exactly when its network can be split like this.

A paper published on 6 May 2013 relies on the same parity fact: one step of its proof uses the fact that a two-sided network has no closed loop of odd length ^[1]. In that paper, the length of a route is the number of lines it uses ^[1], which is the same as counting gear contacts.

The same paper works with a way of adding lines to a two-sided network without breaking the split: a line is added between two dots only when the distance between them is odd and no more than a chosen limit, and the result is still two-sided ^[1].

That gives a handy rule for gear puzzles. If you want to add one contact, or one new gear that touches two existing gears, check the marks first. A new contact between gears with opposite marks is safe. A contact between gears with the same mark creates an odd ring and jams the whole train.

Odd rings can hide inside networks that look tidy. A paper published on 25 October 2012 discusses a well-known small network whose shortest closed loop has five lines ^[2]. Built from gears, that network would jam at once, because its shortest loop has an odd number of gears.

How do belts change the rule?

A belt joins two wheels that do not touch. There are two ways to fit it.

To use belts in the tracing method, write each connection as a pair and note whether it flips or keeps the direction. Meshes and crossed belts flip. Open belts keep. The jam test still works: go round any closed loop and count only the flips. An odd number of flips around a loop means a jam.

How do you work out the speed of a gear?

Meshing gears must have teeth of the same size, and this whole section assumes they do. Where two gears mesh, the same number of teeth passes the contact point on each gear. So a gear with fewer teeth must turn more times to keep up. If a driving gear with forty teeth turns once, a meshing gear with twenty teeth turns twice.

Along a simple chain, the idlers in the middle cancel out. Each one speeds up the motion by as much as it slows it down later. The number of turns of the last gear for each turn of the first is the teeth on the first gear divided by the teeth on the last. Idlers change the direction, but not the final speed.

Be careful with two gears fixed to the same axle. They always turn together at the same speed, so they break the chain into separate parts, and you must work out each part on its own.

How does the method work on a full example?

Here is a worked example with eight gears, A to H, each on its own axle. The contacts are AB, BC, CD, DE, DF, EG, FG and GH. Gear A has a crank and turns clockwise. Which way does gear H turn, and does the train jam?

Step one, look for loops. Gears D, E, G and F form a closed ring, because D touches E and F, and G touches both E and F. That ring has four gears, an even number, so it should not jam. The rest of the network is a plain chain.

Step two, spread outward. A is CW, so B is ACW and C is CW. D is ACW. Both E and F touch D, so both are CW. G touches E and F, and both are CW, so G is ACW without any clash. H touches G, so H is CW.

Step three, check every contact. Every pair on the list has one CW gear and one ACW gear. There is no jam.

Step four, use the shortcut as a check. The shortest route from A to H is A, B, C, D, E, G, H, which uses six contacts. Six is even, so H turns the same way as A. Both methods agree: gear H turns clockwise.

Step five, the speed. Every gear has its own axle, so H turns forty divided by twenty times for each turn of A. Gear H makes two turns for every turn of the crank, whichever way round the ring you follow the motion.

Step six, test a change. Suppose a new gear is added so that it touches both D and E. D is ACW and E is CW, so the new gear would have to be opposite to both. That is impossible, and the whole train would jam. A new gear touching both E and F, which share the mark CW, would simply turn ACW.

A search through every possible choice of direction for all eight gears confirms that exactly one choice obeys every contact, and it is the one in the table.

What mistakes should you avoid?

Can you solve this practice puzzle?

Six toothed gears, A to F, each sit on their own axle. Some axles also carry a belt pulley fixed to the gear, so the pulley and its gear always turn together. Gear A meshes with gear B, and gear B meshes with gear C. A crossed belt joins the pulley on gear C to the pulley on gear D. Gear D meshes with gear E, and an open belt joins the pulley on gear E to the pulley on gear F. Gear A turns clockwise.

First, which way does gear F turn? Second, a seventh gear is to be fitted so that its teeth mesh with exactly two of the gears already there. Which of these four places lets the whole train still turn?

Mark every gear before you look at the options. Try it before reading on.

Answer. Spreading out from A gives A CW, B ACW and C CW. The crossed belt flips, so D is ACW. The mesh flips again, so E is CW, and the open belt keeps the direction, so F turns clockwise.

For the new gear, only the third option works. C and E are both CW, so a gear touching both of them simply turns ACW. Every other option asks the new gear to sit between two gears with opposite marks, which would make it clash with one of them and jam the train. A search through every possible choice of directions confirms that the third option is the only one with a working answer.

Where can you practise next?

Try the same tracing method on this crank puzzle, where the turn of the big gear decides which lantern goes up.

List the contacts, mark the big gear, spread the marks outward and check every contact before you decide which lantern rises.