Six authors tell their stories from their encounters with the famous combinatorial game WYTHOFF NIM and its sequences, including a short survey on exactly covering systems. The volume of the mathematical study of this game is 59% of that of the most ubiquitous game CHESS (MathSciNet). The former originated in 1907, the latter in antiquity. Thus the mathematical study of WYTHOFF NIM may surpass that of CHESS!
1. A modification of the game of NIM
The game of NIM only preceded Wythoff's modification by a few years. By the famous theory of Sprague and Grundy some decades later, NIM drew a lot of attention. WYTHOFF NIM (here also called Wythoff's game), on the other hand, only became regularly revisited towards the end of the 20th century, but its winning strategy is related to Fibonacci's old discovery of the evolution of a rabbit population. The subject has been receiving more attention in recent decades thanks in part to new studies of WYTHOFF NIM and its variants by Fraenkel, and investigations into related sequences and arrays by Kimberling. In this paper we provide surveys of six different aspects of this subject by six of its current masters. Let us recall Wythoff's original definition of the game, given in item 1 in his paper [158]:
The game is played by two persons. Two piles of counters are placed on the table, the number of each pile being arbitrary. The players play alternately and either take from one of the piles an arbitrary number of counters or from both piles an equal number. The player who takes up the last counter or counters, wins.
Wythoff proceeds by designating the safe positions (P-positions in current jargon) of his game, first by noting that the heaps are unordered, which implies
Table 1. The first few terms of the A and B sequences.
that (x, y) is safe if and only if (y, x) is, where x and y denote the respective number of counters in each pile. Then he proceeds to the nowadays celebrated minimal exclusive algorithm (mex), but without giving it a name. Let U be a finite subset of the nonnegative integers. Then the minimal excludant of U, mexU, is the smallest nonnegative integer not in U.
