On the misere version of game Euclid and miserable games

Recently, Gabriel Nivasch got the following remarkable formula for the Sprague–Grundy function of game Euclid: image for all integers image. We consider the corresponding misere game and show that its Sprague–Grundy function image is equal to image for all positions image, except for the case when image or image equals image, where image is the ith Fibonacci number and k is a positive integer. It is easy to see that these exceptional Fibonacci positions are exactly those in which all further moves in the game are forced (unique) and hence, the results of the normal and misere versions are opposite; in other words, for these positions image and image take values 0 and 1 so that image. Let us note that the good old game of Nim has similar property: if there is at most one bean in each pile then all further moves are forced. Hence, in these forced positions the results of the normal and misere versions are opposite, while for all other positions they are the same, as it was proved by Charles Bouton in 1901. Respectively, in the forced positions image and image take values 0 and 1 so that image, where i is the number of non-empty piles, while in all other positions image.