Center for Functional Analysis, Linear Structures and Applications

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Publications

Bounding game temperature using confusion intervals

Publication . Huntemann, Svenja; Nowakowski, Richard; Santos, Carlos

We consider bounds for the temperatures of combinatorial games. Our first result gives an upper bound on the temperatures of the positions of a ruleset in terms of the lengths of the confusion intervals of these positions. We give an example to show that this bound is tight. Our second main result is a method to find a bound for the lengths of the confusion intervals. This pair of results constitutes the first general technique to bound temperatures. As examples of the bound and the method, we consider the temperature of subsets of positions in DOMINEERING and SNORT.

2021-02-06Journal article

Restricted access

Ordinal sums, clockwise hackenbush, and domino shave

Publication . Carvalho, Alda; Huggan, Melissa A.; Nowakowski, Richard; Santos, Carlos

We present two rulesets, domino shave and clockwise hackenbush . The first is somehow natural and, as special cases, includes stirling shave and Hetyei’s Bernoulli game. Clockwise hackenbush seems artificial yet it is equivalent to domino shave. From the pictorial form of the game, and a knowledge of hackenbush, the decomposition into ordinal sums is immediate. The values of clockwise blue-red hackenbush are numbers and we provide an explicit formula for the ordinal sum of numbers where the literal form of the base is { x | } or { | x }, and x is a number. That formula generalizes van Roode’s signed binary number method for blue-red hackenbush.

2020-11-24Journal article

Open access

Combinatorics of JENGA

Publication . Carvalho, Alda; Neto, João; Santos, Carlos

JENGA, a very popular game of physical skill, when played by perfect players, can be seen as a pure combinatorial ruleset. Taking that into account, it is possible to play with more than one tower; a move is made by choosing one of the towers, removing a block from there, that is, a disjunctive sum. JENGA is an impartial combinatorial ruleset, i.e., Left options and Right options are the same for any position and all its followers. In this paper, we illustrate how to determine the Grundy value of a JENGA tower by showing that it may be seen as a bidimensional vector addition game. Also, we propose a class of impartial rulesets, the clock nim games, JENGA being an example of that class.

2020Journal article

Open access