# Task 10.3

Two identical gears with 13 teeth each are placed on top of each other so that the teeth align (so the projection on the plane looks like one 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 on the plane looks like one complete gear? (The gears can be rotated but not flipped.)

## Number of points 7

#
# Answer:

not always.

## Solution

Let's construct a counterexample:

number the teeth in order and remove the teeth numbered $1,2,5,7$.

It is easy to verify that the differences of these numbers give all 13 numbers $0,1,2 \ldots, 12(\bmod 13)$.

Therefore, with each rotation, exactly one pair of "removed teeth" coincides.