  We study the most elementary model of electron motion introduced by R.Feynman
in 1965. It is a game, in which a checker moves on a checkerboard by simple
rules, and we count the turnings. The model is also known as one-dimensional
quantum walk. In his publication, R.Feynman introduces a discrete version of
path integral and poses the problem of computing the limit of the model when
the lattice step and the average velocity tend to zero and time tends to
infinity. We get a nontrivial advance in the problem on the mathematical level
of rigor in a simple particular case; even this case requires methods not known
before. We also prove a conjecture by I.Gaidai-Turlov, T.Kovalev, and A.Lvov on
the limit probability of direction reversal in the model generalizing a recent
result by A.Ustinov.
