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2 results

  • Half-Silvered Mirrors and Wythoff's Game

  • Horacio Porta, Kenneth B. Stolarsky
  • Journal:
    Canadian Mathematical Bulletin / Volume 33 / Issue 1 / 01 March 1990
    Published online by Cambridge University Press:
    20 November 2018, pp. 119-125
    Print publication:
    01 March 1990
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  • We propose the following problem. Given an array of vertical mirrors that simultaneously transmit and reflect, and a single incoming ray of light, describe the configuration of all light rays that are generated. We solve the problem here for a certain infinite configuration of mirrors; the solution involves the winning positions (a(n), b(n)) of Wythoff's game.

  • Beatty Sequences, Continued Fractions, and Certain Shift Operators

  • Kenneth B. Stolarsky
  • Journal:
    Canadian Mathematical Bulletin / Volume 19 / Issue 4 / 01 December 1976
    Published online by Cambridge University Press:
    20 November 2018, pp. 473-482
    Print publication:
    01 December 1976
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  • Let θ = θ(k) be the positive root of θ2 + (k-2)θ-k = 0. Let f(n) = [(n + l)θ]-[nθ] for positive integers n, where [x] denotes the greatest integer in x. Then the elements of the infinite sequence (f(l), f(2), f(3),…) can be rapidly generated from the finite sequence (f(l), f(2),…,f(k)) by means of certain shift operators. For k = 1 we can generate (the characteristic function of) the sequence [nθ] itself in this manner.