80,846 research outputs found

    Blocking Wythoff Nim

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    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≄0k - 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=2k = 2 and k=3k = 3 and, supported by computer simulations, state conjectures of the asymptotic `behavior' of the PP-positions for the respective games when 4≀k≀204 \le k \le 20.Comment: 14 pages, 1 Figur

    Trygg i transport – effekten av tid och upprepning pĂ„ unghĂ€stars lasttrĂ€ning

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    HÀstar transporteras över hela vÀrlden i syften som försÀljning, tÀvling, trÀning och djursjukhusvistelse. I mÄnga fall, men kanske frÀmst vid transport till djursjukhus, hinner aldrig den unga hÀsten fÄ trÀna sig pÄ att lastas innan den ska Äka ivÀg nÄgonstans. Den unga hÀsten utsÀtts för mÄnga obekanta och frÀmmande situationer i samband med lastning, nÄgot som medför att den blir stressad. Den obehagskÀnsla eller stress som hÀsten upplever kan uttryckas pÄ flera olika sÀtt, exempelvis beteendemÀssigt eller fysiologiskt. Det krÀvs dÀrför att hÀsten lÀr sig vad transporten innebÀr och att det inte Àr nÄgot att vara rÀdd för, vilket kan innebÀra en svÄr uppgift för djurhÄllaren. Syftet med den hÀr experimentella studien var att fÄ en uppfattning för hur den unga hÀsten pÄverkas av att lastas samt att ge en förstÄelse för hur den orutinerade hÀstens inlÀrning kan tillÀmpas i vardagliga situationer dÄ hÀsthÄllare ska förbereda transport av sina unga hÀstar. Studien inriktade sig dÀrför pÄ att ta reda pÄ hur hÀsten beter sig vid lastning, hur hÀstens puls förÀndras under lastningstillfÀllet, hur lÄng tid det tar att lasta en unghÀst och hur dessa faktorer förÀndras med antalet gÄnger som hÀsten lastas. I studien anvÀndes sex stycken hÀstar av rasen islandshÀst. Samtliga hÀstar lastades tre gÄnger vardera, en gÄng om dagen i tre dagar. Beteenden som registrerades var fekaliserar, stÄr stilla och drar Ät sidan, det kunde utlÀsas att fler beteenden registrerades under dag 1 Àn under dag 2 och 3. Pulsen ökade markant nÀr hÀstarna gick in i transporten jÀmfört med pulsen som mÀttes innan lastning. Pulsen sjönk betydligt nÀr hÀstarna hade lastats av och stod Äter pÄ fast mark igen. Tiden som det tog att lasta hÀstarna minskade för varje dag som de lastades. Studien visar att unghÀstarna rent fysiologiskt Àr mycket mer pÄverkade av att stÄ inne i transporten Àn vad de Àr nÀr de stÄr pÄ vanlig fast mark. Studien klargör ocksÄ att den unga hÀsten med hjÀlp av inlÀrning förstÄr vad det innebÀr att lastas dÄ resultatet tydliggör en signifikant lÀgre puls men Àven mindre uppvisade beteenden med antalet gÄnger som hÀsten blir lastad. Undantaget för den hÀr studien Àr dock hÀstarna hade högre puls vid andra lastningstillfÀllet Àn vid första. Resultatet frÄn lastningstiden tolkas som att det gÄr fortare att lasta en hÀst pÄ en transport med antalet gÄnger som den blir lastad, Ätminstone om inget har skrÀmt hÀsten under tidigare lastningar. Att lastningstrÀna den unga hÀsten innan transportering anses vara av stort vÀrde för att förbÀttra den unga orutinerade hÀstens vÀlfÀrd 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

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    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 0\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 PP-positions, except 0\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 PP-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 PP-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 (kk-pile Nim)⋆⋆^{\star\star} = kk-pile Nim.Comment: 30 pages, 5 figure

    A Generalized Diagonal Wythoff Nim

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    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 PP-positions are {{0,t}∣t∈N}\{\{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)∣t∈N}\{(t,t)\mid t\in \N\} as options, the new game is Wythoff Nim. It is well-known that the PP-positions of this game lie on two 'beams' originating at the origin with slopes Ί=1+52>1\Phi= \frac{1+\sqrt{5}}{2}>1 and 1Ί<1\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 PP-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 PP-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}\{pt, qt\}, where 0<p<q0 < p < q are fixed positive integers and t>0t > 0? Does the answer perhaps depend on the specific values of pp and qq? We state three conjectures of which the weakest form is: lim⁥t∈Nbtat\lim_{t\in \N}\frac{b_t}{a_t} exists, and equals Ί\Phi, if and only if (p,q)(p, q) is a certain \emph{non-splitting pair}, and where {{at,bt}}\{\{a_t, b_t\}\} represents the set of PP-positions of the new game. Then we prove this conjecture for the special case (p,q)=(1,2)(p,q) = (1,2) (a \emph{splitting pair}). We prove the other direction whenever q/p<Ίq / 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
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