Folkpor

a) Two identical gears with 14 teeth each are placed one on top of the other so that the teeth align (so the projection onto a plane looks like a single gear). After this, four pairs of aligned teeth are cut out. Is it always possible to rotate these gears relative to each other so that the projection onto a plane looks like a single complete gear? (The gears can be rotated but not flipped.)

b) The same question for two gears with 13 teeth, from which 4 teeth have been cut out.
a) In the initial position, four pairs of cut-out teeth coincide (one tooth on the first gear and one on the second). There are 16 such pairs in total. Therefore, in one of the 13 remaining positions, no pair coincides.

b) Counterexample: number the teeth in order and cut out the teeth with numbers $1, 2, 5, 7$. As is easily verified, the differences of these numbers give all 13 numbers

$0, 1, 2 \ldots, 12 (\bmod 13)$. Therefore, with each rotation, exactly one pair of "cut-out teeth" coincides.

## Answer

a) Always b) not always