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| 2. |
- Hegarty, Peter, 1971, et al.
(författare)
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The structure of maximum subsets of {1,...,n} with no solutions to a+b=kc
- 2005
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Ingår i: Electron. J. Combin.. ; 12
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Tidskriftsartikel (refereegranskat)abstract
- If $k$ is a positive integer, we say that a set $A$ of positive integers is $k$-sum-free if there do not exist $a,b,c$ in $A$ such that $a + b = kc$. In particular we give a precise characterization of the structure of maximum sized $k$-sum-free sets in $\{1,...,n\}$ for $k\ge 4$ and $n$ large.
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| 3. |
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| 4. |
- Larsson, Urban, 1965
(författare)
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2-Pile Nim With a Restricted number of Move-size Imitations (with an appendix by Peter Hegarty)
- 2009
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Ingår i: Integers Journal. ; 9:G4, s. 671-690
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Tidskriftsartikel (refereegranskat)abstract
- We study a variation of the combinatorial game of 2-pile Nim. Move as in 2- pile Nim but with the following constraint: Suppose the previous player has just removed say x > 0 tokens from the shorter pile (either pile in case they have the same height). If the next player now removes x tokens from the larger pile, then he imitates his opponent. For a predetermined natural number p, by the rules of the game, neither player is allowed to imitate his opponent on more than p−1 consecutive moves. We prove that the strategy of this game resembles closely that of a variant of Wythoff Nim—a variant with a blocking manoeuvre on p − 1 diagonal positions. In fact, we show a slightly more general result in which we have relaxed the notion of what an imitation is. The paper includes an appendix by Peter Hegarty, Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, hegarty@chalmers.se.
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| 5. |
- Larsson, Urban, 1965
(författare)
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A Generalized Diagonal Wythoff Nim
- 2010
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- In this paper we study a family of 2-pile take away games, that we denote by Generalized Diagonal Wythoff Nim (GDWN).The story begins with 2-pile Nim whose sets of moves and $P$-positions are $\{\{0,t\}\mid t\in \N\}$ and $\{(t,t)\mid t\in \M \}$ respectively. If we adjoin to 2-pile Nim the main-\emph{diagonal} $\{(t,t)\mid t\in \N\}$ as moves, the new game is Wythoff Nim. It is well-known that the $P$-positions of this game lie on two 'beams' starting at the origin with slopes $\phi = \frac{1+\sqrt{5}}{2}>1$ and $\frac{1}{\phi } < 1$. Hence one may think of this as if, in the process of going from Nim to Wythoff Nim, the set of $P$-positions has \emph{split} and landed some distance off the main diagonal. This geometrical observation has motivated us to ask the following intuitive question. Does this splitting of the set of $P$-positions continue in some meaningful way if we adjoin to the game of Wythoff Nim new \emph{generalized diagonal} moves of the form $(pt, qt)$ and $(qt, pt)$, where $p < q$ are fixed positive integers and $t$ ranges over the positive integers? Does the answer depend on the specific values of $p$ and $q$? We state three conjectures of which the weakest form is: $\lim_{t\in \N}\frac{b_t}{a_t}$ exists, and equals $\phi$, if and only if $(p, q)$ is a certain \emph{non-splitting pair}, and where $\{(a_t, b_t),(b_t,a_t)\}$ represents the set of $P$-positions of the new game. Then we prove this conjecture for the special case $(p,q) = (1,2)$ (a \emph{splitting pair}). We prove the other direction whenever $q / p < \phi$. A variety of experimental data is included, aiming to point out some directions for future work on GDWN games.
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| 6. |
- Larsson, Urban, 1965
(författare)
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A Generalized Diagonal Wythoff Nim
- 2012
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Ingår i: Integers. - 1553-1732. ; 12:G2, s. 1-24
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Tidskriftsartikel (refereegranskat)abstract
- The P-positions of the 2-pile take-away game of Wythoff Nim lie on two beams of slope (sqrt(5)+1)/2 and (sqrt(5)−1)/2 respectively. We study extensions to this game where a player may also remove simultaneously pt tokens from either of the piles and qt from the other, where p < q are given positive integers and where t ranges over the positive integers. We prove that for certain pairs (p, q) the P-positions are identical to those of Wythoff Nim, but for (p, q) = (1, 2) they do not even lie on two beams. By several experimental results, we conjecture a classification of all pairs (p, q) for which Wythoff Nim’s beams of P-positions transform via a certain splitting behavior, similar to that of going from 2-pile Nim to Wythoff Nim.
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| 7. |
- Larsson, Urban, 1965
(författare)
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A Generalized Diagonal Wythoff Nim
- 2012
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Ingår i: Integers. - 1553-1732. ; 12:G2, s. 1-24
-
Tidskriftsartikel (refereegranskat)abstract
- The P-positions of the 2-pile take-away game of Wythoff Nim lie on two beams of slope (sqrt(5)+1)/2 and (sqrt(5)−1)/2 respectively. We study extensions to this game where aplayer may also remove simultaneously pt tokens from either of the piles and qt from the other, where p
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| 8. |
- Larsson, Urban, 1965
(författare)
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A Generalized Diagonal Wythoff Nim
- 2010
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- In this paper we study a family of 2-pile take away games, that we denote by Generalized Diagonal Wythoff Nim (GDWN).The story begins with 2-pile Nim whose sets of moves and $P$-positions are $\{\{0,t\}\mid t\in \N\}$ and $\{(t,t)\mid t\in \M \}$ respectively. If we adjoin to 2-pile Nim the main-\emph{diagonal} $\{(t,t)\mid t\in \N\}$ as moves, the new game is Wythoff Nim. It is well-known that the $P$-positions of this game lie on two 'beams' starting at the origin with slopes $\phi = \frac{1+\sqrt{5}}{2}>1$ and $\frac{1}{\phi }
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| 9. |
- Larsson, Urban, 1965
(författare)
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Blocking Wythoff Nim
- 2010
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- The 2-player impartial game of Wythoff Nim is played on two piles of tokens. A move consists in removing any number of tokens from precisely one of the piles or the same number of tokens from both piles. The winner is the player who removes the last token. We study this game with a blocking maneuver, that is, for each move, before the next player moves the previous player may declare at most a predetermined number, $k - 1 \ge 0$, of the options as forbidden. When the next player has moved, any blocking maneuver is forgotten and does not have any further impact on the game. We resolve the winning strategy of this game for $k = 2$ and $k = 3$ and, supported by computer simulations, state conjectures of the asymptotic behavior of the $P$-positions for the respective games when $4 \le k \le 20$.
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| 10. |
- Larsson, Urban, 1965
(författare)
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Blocking Wythoff Nim
- 2011
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Ingår i: The Electronic Journal of Combinatorics. - : The Electronic Journal of Combinatorics. - 1077-8926 .- 1097-1440. ; 18:1
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Tidskriftsartikel (refereegranskat)abstract
- The 2-player impartial game of Wythoff Nim is played on two piles of tokens. A move consists in removing any number of tokens from precisely one of the piles or the same number of tokens from both piles. The winner is the player who removes the last token. We study this game with a blocking maneuver, that is, for each move, before the next player moves the previous player may declare at most a predetermined number, k − 1 ≥ 0, of the options as forbidden. When the next player has moved, any blocking maneuver is forgotten and does not have any further impact on the game. We resolve the winning strategy of this game for k = 2 and k = 3 and, supported by computer simulations, state conjectures of ‘sets of aggregation points’ for the P-positions whenever 4 ≤ k ≤ 20. Certain comply variations of impartial games are also discussed.
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