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features The Optimal Dartboard?

By David F. Percy Cmath Fima

Abstract

Our challenge is to determine the best arrangement of the numbers 1, 2,...,20 on a dartboard. This problem has been tackled before but we consider a new constraint and a different optimality criterion that lead to an original solution. 1 Problem definition There are 20! ≈ 2 × 10 18 possible arrangements of the numbers 1, 2,...,20 on a dartboard and 19!/2 ≈ 6 × 10 16 distinct cycles that allow for reflection and rotation. Our aim is to determine a cycle that is optimal in some sense. There are three constraints that we wish to impose on any cycle as follows. 1. Penalise mistakes by overambitious players. This was apparent in the standard dartboard designed by Brian Gamlin in 1896, in which large numbers tend to be adjacent to small numbers. 2. Alternate odd and even numbers. This parity criterion was proposed by Eastaway and Haigh [1]. It is particularly appealing because it induces a degree of symmetry and ensures a challenging endgame. 3. Exhibit rotational quasi-symmetry. We propose this criterion to ensure that similar clusters of adjacent sectors all around the dartboard offer similar rewards to players. The notation that we adopt is as follows. Define the twenty numbers reading clockwise from the top of a dartboard to be xi for i =1, 2,...,20. The ordered set (x1,x2,...,x20) forms a cycle and we define x0 = x20 for convenience. The standard dartboard has the arrangemen

Year: 2013
OAI identifier: oai:CiteSeerX.psu:10.1.1.352.5469
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