  We find conditions under which the P-positions of three subtraction games
arise as pairs of complementary Beatty sequences. The first game is due to
Fraenkel and the second is an extension of the first game to non-monotone
settings. We show that the P-positions of the second game can be inferred from
the recurrence of Fraenkel's paper if a certain inequality is satisfied. This
inequality is shown to be necessary if the P-positions are known to be pairs of
complementary Beatty sequences, and the family of irrationals for which this
inequality holds is explicitly given. We highlight several games in the
literature that have P-positions as pairs of complementary Beatty sequences
with slope in this family. The third game we present is novel, and we show that
the P-positions can be inferred from the same recurrence in any setting. It is
shown that any pair of complementary Beatty sequences arises as the P-positions
of some game in this family. We also provide background on some inverse
problems which have appeared in the field over the last several years, in
particular the Duch\^ene-Rigo conjecture. This paper presents a solution to the
Fraenkel problem posed at the 2011 BIRS workshop, a modification of the
Duch\^ene-Rigo conjecture.
