17.5 Wythoff’s game 513
Appendix: Beatty’s Theorem
Ifαandβare positive irrational numbers satisfyingα^1 +^1 β=1, then the
sequences
α, 2 α, 3 α, ...
and
β, 2 β, 3 β, ...
form a partition of the sequence of positive integers.
See Russian problem book, MG Problem 1300, solution, 62 (1989)
203.
Ifrandsare positive integers andn=r+s, then the union of the
two sequences
⌊nr⌋
,
⌊
2 n
r⌋
, ...,
⌊
(r−1)n
r⌋
,
and ⌊
n
s
⌋
,
⌊
2 n
s⌋
, ...,
⌊
(s−1)n
s⌋
.
is the natural sequence 1 , 2 , ...,n− 2 if and only ifrandsare relatively
prime.