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Picture of How to Build a Lego(R) Chainsaw
Having seen a 'chainsaw' instructable which was not a chainsaw, I started thinking. I had some Lego bits, including:

A motor (4.5v)
Some gear wheels
Some chain links
And some other general bits.

These parts are required, as pictured.

The tricky part was the saw teeth...
 
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Step 1: Saw Teeth

These are the most tricky part, which is why I started on them first.

If you examine Lego chain-links, you will see that they interlock with pins spaced 6.5mm apart. There's enough clearence to get something very thin between the joints to be fixed on the same pins:

Aluminium beer-can, easily cut with scissors & sharp knives (do be careful).

Tools:
A hammer
A knife blade
A piece of stiff wire (from an umbrella)
A picture nail
An abrasive stick (coarse corundum, probably)
A piece of wood

Please view the pictures for the method.

Step 2: Fitting the teeth

Picture of Fitting the teeth
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It's best that you work this out as you do this.
Mount one tooth on one pin of a chain link, and secure it by clipping another link over the top.
Then mount the other hole on the tooth on a third link, and clip it to the first, securing the tooth.
(If anyone gets this far, and I doubt it, they'll work it out)

Step 3: And the rest

Picture of And the rest
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I'm not going into details, this part is relatively easy.
The pictures should give you enough information.
I started with a direct-drive from the motor pinion, but later geared-down (intro picture). However, direct-drive off the motor works best.

Step 4: Action

Picture of Action
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Some paper was shredded, but there was some slippage where the steel motor-shaft should grip the black plastic pinion.
Weedy, but concept proven.

Some glue on the motor pinion and this was pretty effective on broccoli.
Success!