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]:

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.