The Sprague–Grundy function of the real game Euclid

The game Euclid, introduced and named by Cole and Davie, is played with a pair of nonnegative integers. The two players move alternately, each subtracting a positive integer multiple of one of the integers from the other integer without making the result negative. The player who reduces one of the integers to zero wins. Unfortunately, the name Euclid   has also been used for a subtle variation of this game due to Grossman in which the game stops when the two entries are equal. For that game, Straffin showed that the losing positions (a,b)(a,b) with a