Blocking Wythoff Nim

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 $4 \le k \le 20$.Comment: 14 pages, 1 Figur

Trygg i transport – effekten av tid och upprepning på unghästars lastträning

Hästar transporteras över hela världen i syften som försäljning, tävling, träning och djursjukhusvistelse. I många fall, men kanske främst vid transport till djursjukhus, hinner aldrig den unga hästen få träna sig på att lastas innan den ska åka iväg någonstans. Den unga hästen utsätts för många obekanta och främmande situationer i samband med lastning, något som medför att den blir stressad. Den obehagskänsla eller stress som hästen upplever kan uttryckas på flera olika sätt, exempelvis beteendemässigt eller fysiologiskt. Det krävs därför att hästen lär sig vad transporten innebär och att det inte är något att vara rädd för, vilket kan innebära en svår uppgift för djurhållaren. Syftet med den här experimentella studien var att få en uppfattning för hur den unga hästen påverkas av att lastas samt att ge en förståelse för hur den orutinerade hästens inlärning kan tillämpas i vardagliga situationer då hästhållare ska förbereda transport av sina unga hästar. Studien inriktade sig därför på att ta reda på hur hästen beter sig vid lastning, hur hästens puls förändras under lastningstillfället, hur lång tid det tar att lasta en unghäst och hur dessa faktorer förändras med antalet gånger som hästen lastas. I studien användes sex stycken hästar av rasen islandshäst. Samtliga hästar lastades tre gånger vardera, en gång om dagen i tre dagar. Beteenden som registrerades var fekaliserar, står stilla och drar åt sidan, det kunde utläsas att fler beteenden registrerades under dag 1 än under dag 2 och 3. Pulsen ökade markant när hästarna gick in i transporten jämfört med pulsen som mättes innan lastning. Pulsen sjönk betydligt när hästarna hade lastats av och stod åter på fast mark igen. Tiden som det tog att lasta hästarna minskade för varje dag som de lastades. Studien visar att unghästarna rent fysiologiskt är mycket mer påverkade av att stå inne i transporten än vad de är när de står på vanlig fast mark. Studien klargör också att den unga hästen med hjälp av inlärning förstår vad det innebär att lastas då resultatet tydliggör en signifikant lägre puls men även mindre uppvisade beteenden med antalet gånger som hästen blir lastad. Undantaget för den här studien är dock hästarna hade högre puls vid andra lastningstillfället än vid första. Resultatet från lastningstiden tolkas som att det går fortare att lasta en häst på en transport med antalet gånger som den blir lastad, åtminstone om inget har skrämt hästen under tidigare lastningar. Att lastningsträna den unga hästen innan transportering anses vara av stort värde för att förbättra den unga orutinerade hästens välfärd vid transport.Horses are transported around the world for purposes such as sales, competition, training and veterinary care. In many cases, mainly in transport to the veterinary hospital, the horse has not been trained to be loaded before being transported. The young horse is exposed to many unfamiliar and potentially frightening situations during loading, which can result in stress. The discomfort or stress the horse is experiencing can be expressed behaviorally and/or physiologically. It is therefore required that the horse is habituated to the vehicle and transportation, which could be a difficult task for the horse owner. The aim of this study was to achieve an understanding of how the young horse is affected by loading. I also wanted to show how the inexperienced horse learning theory can be applied in everyday situations when horse keepers prepare for transporting their young horses. The study focused therefore on how the horse behaves when loading, how the horse's heart rate changes during the time of loading, how long it takes to load a young horse, and how these factors change with the number of times the horse is loaded. The study used six Icelandic horses. All horses were loaded three times each, once a day for three consecutive days. The results of the heart rate and loading time were compared and tested for statistical significance. Behaviors recorded were defecation, stand calm and pull to one side. It could be deduced that more behaviors were recorded on day 1 than on day 2 and 3. Heart rate was significantly higher when the horses were in the trailer (P= 0.042) compared with the heart rate that was measured before loading. Heart rate decreased significantly when the horses were unloaded and stood back on solid ground again (P= 0.008). The time it took to load the horses decreased significantly by day (P= 0.002). The study shows that young horses, physiologically, is significantly affected by standing inside the vehicle compared to when they are outside on solid ground. The study also clarifies that the young horse with the help of learning, through repeated exposure, understands what it means to be loaded as the result elucidates a significantly lower heart rate, but also less behavior exhibited by the number of times that the horse will be loaded. Somewhat surprising was that the horses had a higher heart rate the second time of loading than the first. Loading time was reduced with an increasing number of times the horse was loaded. However, this effect might be reversed if something scares the horse during loading. To train the young horse of loading before transportation is considered to be of great value to improve the young horse welfare and human safety during transport

The ⋆\star-operator and Invariant Subtraction Games

We study 2-player impartial games, so called \emph{invariant subtraction games}, of the type, given a set of allowed moves the players take turn in moving one single piece on a large Chess board towards the position $\boldsymbol 0$. Here, invariance means that each allowed move is available inside the whole board. Then we define a new game, $\star$ of the old game, by taking the $P$-positions, except $\boldsymbol 0$, as moves in the new game. One such game is \W^\star= (Wythoff Nim)$^\star$, where the moves are defined by complementary Beatty sequences with irrational moduli. Here we give a polynomial time algorithm for infinitely many $P$-positions of \W^\star. A repeated application of $\star$ turns out to give especially nice properties for a certain subfamily of the invariant subtraction games, the \emph{permutation games}, which we introduce here. We also introduce the family of \emph{ornament games}, whose $P$-positions define complementary Beatty sequences with rational moduli---hence related to A. S. Fraenkel's `variant' Rat- and Mouse games---and give closed forms for the moves of such games. We also prove that ($k$-pile Nim)$^{\star\star}$ = $k$-pile Nim.Comment: 30 pages, 5 figure

A Generalized Diagonal Wythoff Nim

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 options and $P$-positions are $\{\{0,t\}\mid t\in \N\}$ and \{(t,t)\mid t\in \M \} respectively. If we to 2-pile Nim adjoin the main-\emph{diagonal} $\{(t,t)\mid t\in \N\}$ as options, the new game is Wythoff Nim. It is well-known that the $P$-positions of this game lie on two 'beams' originating 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, to the game of Wythoff Nim, adjoin some new \emph{generalized diagonal} move, that is a move of the form $\{pt, qt\}$, where $0 < p < q$ are fixed positive integers and $t > 0$? Does the answer perhaps 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 $\{\{a_t, b_t\}\}$ 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 $q / p < \Phi$. In the Appendix, a variety of experimental data is included, aiming to point out some directions for future work on GDWN games.Comment: 38 pages, 34 figure