Posts Categorized: games

  • games, number theory

    Life on Gaussian primes

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    At the moment I’m re-reading Siobhan Roberts’ biography of John Horton Conway, Genius at play – the curious mind of John Horton Conway. In fact, I’m also re-reading Alexander Masters’ biography of Simon Norton, The genius in my basement – the biography of a happy man. [full_width_image] [/full_width_image] If you’re in for a suggestion, try… Read more »

  • games, geometry

    The geometry of football

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    Soon, we will be teaching computational geometry courses to football commentators. If a player is going to be substituted we’ll hear sentences like: “no surprise he’s being replaced, his Voronoi cell has been shrinking since the beginning of the second half!” David Sumpter, the author of Soccermatics: Mathematical Adventures in the Beautiful Game, wrote a… Read more »

  • books, games

    human-, computer- and fairy-chess

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    It was fun following the second game last night in real time. Carlsen got a winning endgame with two bishops against a rook, but blundered with 62. Bg4?? (winning was Kf7), resulting in stalemate. There was this hilarious message around move 60: “The computer has just announced that white mates in 31 moves. Of course,… Read more »

  • games

    How to win transfinite Nimbers?

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    Last time we introduced the game of transfinite Nimbers and asked a winning move for the transfinite game with stones a at position $~(2,2) $, b at $~(4,\omega) $, c at $~(\omega+2,\omega+3) $ and d at position $~(\omega+4,\omega+1) $. Above is the unique winning move : we remove stone d and by the rectangle-rule add… Read more »

  • games

    n-dimensional and transfinite Nimbers

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    Today, we will expand the game of Nimbers to higher dimensions and do some transfinite Nimber hacking. In our identification between $\mathbb{F}_{16}^* $ and 15-th roots of unity, the number 8 corresponds to $\mu^6 $, whence $\sqrt{8}=\mu^3=14 $. So, if we add a stone at the diagonal position (14,14) to the Nimbers-position of last time… Read more »

  • games

    How to play Nimbers?

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    Nimbers is a 2-person game, winnable only if you understand the arithmetic of the finite fields $\mathbb{F}_{2^{2^n}} $ associated to Fermat 2-powers. It is played on a rectangular array (say a portion of a Go-board, for practical purposes) having a finite number of stones at distinct intersections. Here’s a typical position The players alternate making… Read more »

  • games, number theory

    Seating the first few thousand Knights

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    The Knight-seating problems asks for a consistent placing of n-th Knight at an odd root of unity, compatible with the two different realizations of the algebraic closure of the field with two elements.

  • games, groups

    can -oids save group-theory 101?

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    Can one base a group-theory 101 course on the notion of groupoids?

  • games, number theory

    On2 : extending Lenstra’s list

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    Hendrik Lenstra found an effective procedure to compute the mysterious elements alpha(p) needed to do actual calculations with infinite nim-arithmetic.

  • featured, games, number theory

    On2 : Conway’s nim-arithmetics

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    Conway’s nim-arithmetic on ordinal numbers leads to many surprising identities, for example who would have thought that the third power of the first infinite ordinal equals 2…

  • games, number theory

    On2 : transfinite number hacking

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    Surely Georg Cantor’s transfinite ordinal numbers do not have a real-life importance? Well, think again.

  • games, groups

    sporadic simple games

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    About a year ago I did a series of posts on games associated to the Mathieu sporadic group $M_{12} $, starting with a post on Conway’s puzzle M(13), and, continuing with a discussion of mathematical blackjack. The idea at the time was to write a book for a general audience, as discussed at the start… Read more »

  • games

    Surreal numbers & chess

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    Most chess programs are able to give a numerical evaluation of a position. For example, the position below is considered to be worth +8.7 with white to move, and, -0.7 with black to move (by a certain program). But, if one applies combinatorial game theory as in John Conway’s ONAG and the Berlekamp-Conway-Guy masterpiece Winning… Read more »