Not being an automotive design guy in any shape or form, my first guess... does it have anything to do with directly opposing cylinders firing at the same time being a bad thing?
That is a very interesting question. All the cylinders (effectively) operate off a single crankpin. So all the cylinders must fire in sequence around the engine. But each cylinder only fires every other revolution.
I guess the firing sequence would be something like 1-3-5-7-9-2-4-6-8. Every second cylinder in sequence fires around the engine. Pairs of side by side cylinders would reach TDC almost simultaneously. one would be at TDC firing, and the other at TDC overlap. That sort of sequence only works with odd numbers of cylinders.
MOST radial piston engines have an odd number of cylinders per crankpin because they are four- strokes, and commonly use a shared cam ring running at half output speed to actuate the valves on all cylinders. With a little mental gyration, you can eventually come to the understanding that there must be an even number of cylinders, or you run out of cylinders after one cam revolution.
It is not accurate to say that ALL radial four strokes must have an odd number of cylinders per row. I am aware of two exceptions, both produced early in the 20th Century. The Fairchild- Caminez had four cylinders, and the Marchetti had eight. I think the Fairchild- Caminez was produced in some quantity, and there may have been variants. The Marchetti was produced not by the Italian airplane manufacturer, but by Paul Marchetti of San Francisco, who died while flight testing an airframe in which to flight test his engine.
I got interested when I saw a photo of what must be the Marchetti engine, sitting alone somewhere, in some technical fish wrapper, with a derisive caption about how stupid you had to be to actually build a radial eight.
I can't remember which fish wrapper, but I did find images from Paul's advertising campaign, which you can probably Google up.
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There is a trick, of course. Both those odd engines use sturdy desmodromic cams and rockers to actuate the pistons. There is no crankpin. Each cylinder does a complete cycle in one output revolution, so the 'crank cams' are symmetrical and naturally balanced.
It uses a supercharger and a turbocharger and uses a compressed air bottle as a starter when it dumps air into the direct drive supercharger to turn the engine over to start it.
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I have been called "A storehouse of worthless information" many times.