Richard K. Guy

The G-values of various games

A disjunctive combination of a finite set of two-person games Γ 1 , Γ 2 , …, Γ k may be defined thus: The players play alternately, each in turn making a move in one and only one of the individual games. If, in addition, the conditions are imposed that (i) a player loses if unable to move (in any game), (ii) the games are impartial , i.e. the allowable moves from any position do not depend on which player is about to play (or on the previous moves, though these can be ‘built in’ to the position if necessary), (iii) the games are of bounded play , i.e. for each game Γ i corresponding to any initial position P j there is an integer b ij such that the game must terminate after at most b ij moves, then Grundy (6) has shown that there is a function G ( P ) (called by him Ω( P )) of the positions P with the following properties: (1 a ) G(P) = 0 for a terminal position, from which no move is possible; for other positions G(P) is the smallest non-negative integer different from all values of G(Q i ) , where there is a permissible move from P to Q i , (1 b ) for a disjunctive combination of games, G for the combined position is the nim-sum of the G s the individual positions. By the nim-sum, we mean that each separate G is to be written in the scale of 2, as Σ a r 2 r , and then in forming the sum, the a r s are to be added mod 2 for each value of r , as in the theory of Nim((1), (7), (8), pp. 36–8).

Unsolved problems in Combinatorial Games

We have retained the numbering from the list of unsolved problems given on pp. 183–189 of Amer. Math. Soc. Proc. Sympos. Appl Math. 43(1991) and added in some new material. For many more references than we list, see Fraenkel’s Bibliography.

The toroidal crossing number of the complete graph

Abstract The toroidal crossing number of the complete graph on n points is shown to lie between 23 210 ( n 4 ) and 59 216 ( n − 1 4 ) provided n ≥10. For n =8, 9 and 10 the result is 4, 9, and 23, respectively.

Computational Aesthetics 2008: Automatically mimicking unique hand-drawn pencil lines

In applications such as architecture, early design sketches containing accurate line drawings often mislead the target audience. Approximate human-drawn sketches are typically accepted as a better way of demonstrating fundamental design concepts. To this end we have designed an algorithm that creates lines that perceptually resemble human-drawn lines. Our algorithm works directly with input point data and a physically based mathematical model of human arm movement. Our algorithm generates unique lines of arbitrary length given the end points of a line, without relying on a database of human-drawn lines. We found that an observational analysis obtained through various user studies of human lines made a bigger impact on the algorithm than a statistical analysis. Additional studies have shown that the algorithm produces lines that are perceptually indistinguishable from that of a hand-drawn straight pencil line. A further expansion to the system resulted in mimicked dashed lines.

Equitable and proportional coloring of trees

Abstract We show that almost all trees can be equitably 3-colored, that is, with three color classes of cardinalities differing by at most one. Also, except in some extreme cases, they can be 3-colored with color classes of sizes in given proportions.

The toroidal crossing number of Km,n

Abstract It is shown that the toroidal crossing number of the complete bipartite graph, Km, n, lies between 1 15 ( m 2 ) ( n 2 ) and 1 6 ( m − 1 2 ) ( n − 1 2 ) the lower bound holding for sufficiently large m and n.

Sets of Integers Whose Subsets Have Distinct Sums

Publisher Summary This chapter presents a study on sets of integers, whose subsets have distinct sums. The subsets of the set of integers {2 i : 0 ≤ i ≤ k ) all have distinct sums. For a maximum number, m , of positive integers, the chapter discusses if it is possible to have m = k + 2, when x = 2 k . If a 1 a 2 m are positive integers whose subsets have distinct sums, then a i ≥ 2 m – 1. The chapter also discusses an upper bound for m, the Conway–Guy sequence, sets of numbers with distinct sums of subsets and presents several theorems and lemmas.

MONTHLY unsolved problems, 1969-1997

In this department the MONTHLY presents easily stated unsolued problems dealing with notions ordinarily encountered in undergraduate mathematics. Each problem should be accompanied by relevant references (if any are known to the author) and by a brcef description of known partial or related resrlts. Typescripts should be sent to Richard Nowakowski, Department of Mathematics & Statistics & Comprting Science, Dalhousie University, Halifax NS, Canada B3H 3J5, ryn@cs.dal.ca

Which integers are representable as the product of the sum of three integers with the sum of their reciprocals

For example, the integer 564 is so representable by the three integers 122 44200 5010002877 81163 51171 17995 2136135134 91867, -3460 69586 84255 04865 64589 22621 88752 08971 30654 24460 and 74807 19101 53025 27837 94583 60171 46464 94820 59055 28060.

Primes at a glance

Let N = B L, B > ILI, gcd(B, L) = 1, p I BL for all primes p < N. Then N is 0, 1 or a prime. Writing N in this form suggests a primality and a squarefreeness test. If we also require that when the prime q I BL and p < q then p I BL, we say that B L is a presentation of N. We list all presentations found for any N. We believe our list is complete.