For gears to work, one gear has to turn one way and the other the opposite way. If there’s three, five, seven, or any other odd number of gears, one of them will end up interlocking with a gear going the wrong direction.
In the interesting Numberphile video, mathematician Henry Segerman talks about 3 gear systems and how, despite their seeming impossibility, can actually exist, with the help of a little ingenuity and 3D printing.
