Neil A. McKay

Misère-play Hackenbush Sprigs

A Hackenbush Sprig is a Hackenbush string with the ground edge coloured green and the remaining edges either red or blue. We show that in canonical form a Sprig is a star-based number (the ordinal sum of star and a dyadic rational) in misère-play, as well as in normal-play. We find the outcome of a disjunctive sum of Sprigs in misère-play and show that it is the same as the outcome of that sum plus star in normal-play. Along the way it is shown that the sum of a Sprig and its negative is equivalent to 0 in the universe of misère-play dicots, answering a question of Allen.

Wythoff partizan subtraction

We introduce a class of normal-play partizan games, called Complementary Subtraction. These games are instances of Partizan Subtraction where we take any set A of positive integers to be Left’s subtraction set and let its complement be Right’s subtraction set. In wythoff partizan subtraction we take the set A and its complement B from wythoff nim, as the two subtraction sets. As a function of the heap size, the maximum size of the canonical forms grows quickly. However, the value of the heap is either a number or, in reduced canonical form, a switch. We find the switches by using properties of the Fibonacci word and standard Fibonacci representations of integers. Moreover, these switches are invariant under shifts by certain Fibonacci numbers. The values that are numbers, however, are distinct, and we can find their binary representation in polynomial time using a representation of integers as sums of Fibonacci numbers, known as the ternary (or “the even”) Fibonacci representation.

Canonical forms of uptimals

Every all-small game has mean 0 and temperature 0. Therefore, the temperature theory is of no use in the study of all-small games. The main tool used in analyzing all-small games since the 1970s has been approximation by atomic weight. In the 1980s, Conway and Ryba developed (but did not publish) the uptimal theory, which is finer and more precise than the atomic weight theory. In order to study games such as clobber and push-ups, the author independently advances the theory of uptimals. In particular, the author finds the canonicals forms of all integral uptimals. The canonical forms also lead us to an algorithm for recognizing uptimals. We end with examples of rulesets with non-trivial uptimal values.