Checkers, also called draughts, is a family of two-player games played on a checkerboard. English
draughts, also called American checkers, is played on an 
checkerboard with
initially twelve pieces per player (World Checkers/Draughts Federation). International
draughts is played on a 
checkerboard with initially twenty pieces per player
(World Draughts Federation). The board size and rules therefore
depend on the variant.
Pool checkers, also called Spanish pool checkers, is a distinct 
variant. Unlike English
draughts, it allows uncrowned pieces to capture backwards and crowned pieces
to move any distance along an unobstructed diagonal (Olsen 2001; Ratrout 2018, p. 12).
Checkers sets commonly have black and red boards and pieces. The official colors depend on the variant and its governing federation.
For 
or 4 and a positive
integer 
, let 
be the directed graph
whose vertices are 
,
with coordinate indices reduced by modular arithmetic
modulo 
and 
,
and whose directed edges are 
and 
. It is the graph Cartesian product of two directed cycles. When 
, the second directed graph
edge at each graph vertex is a graph
loop. The numbers 1, 40, 793, 12800, 193721, ... (OEIS A006240)
and 1, 13, 108, 793, 5611, 39312, ... (OEIS A006239)
count Eulerian circuits beginning with one specified
directed graph edge in 
and 
, respectively, for 
, 2, .... They do not count two different conventions
for circuits on the same square checkerboard.
