Joint landscape on a peg solitaire board

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Niveau: Supérieur, Doctorat, Bac+8
Joint landscape on a peg solitaire board?† O. Ramaré, CNRS, Laboratoire Painlevé, Université Lille 1 59 655 Villeneuve d'Ascq, France May 31, 2011 Abstract We investigate and develop further the notions of joint landscape. 1 Introduction We introduced in (Ramaré, 2008c) the notion of landscape for a position I and in (Ramaré, 2008a) (see also the end of (Ramaré, 2008b)) the notion of joint landscape for two positions I and J . It is the function L(A, I, J) on S which is equal in A, when A /? I, to the minimal number of moves required to reach a position from which we may still derive J and which contains A, when starting from I. We have called this number the joint height h(A, I, J) of A with respect to be I and J , When A ? I, then ?L(A, I, J) is the minimal number of moves required to reach a position from which we may still derive J and which does not contain A. We call this number the joint depth of A with respect to I and J , an easy specialisation of the notion of depth introduced in section 11 of (Ramaré, 2008c). These heights, resp.

  • whenever no

  • final position

  • solitaire board

  • path verify

  • height functions

  • peg solitaire

  • f1 ·

  • external height

  • lesser external height


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∗† Joint landscape on a peg solitaire board
O. RamarÉ, CNRS, Laboratoire PainlevÉ, UniversitÉ Lille 1 59 655 Villeneuve d’Ascq, France May 31, 2011
Abstract We investigate and develop further the notions of joint landscape.
1 Introduction We introduced in (RamarÉ, 2008c) the notion oflandscapefor a positionI and in (RamarÉ, 2008a) (see also the end of (RamarÉ, 2008b)) the notion of joint landscapefor two positionsIandJ. Itis the functionL(A, I, J)onS which is equal inA, whenA/ I, to the minimal number of moves required to reach a position from which we may still deriveJand which containsA, when starting fromI. Wehave called this number thejoint heighth(A, I, J)ofA with respect to beIandJ, WhenAI, thenL(A, I, J)is the minimal number of moves required to reach a position from which we may still derive Jand which does not containA. Wecall this number thejoint depthof Awith respect toIandJ, an easy specialisation of the notion of depth introduced in section 11 of (RamarÉ, 2008c).These heights, resp.depths, in Aare infinite when it is not possible to put a peg inAnot possible to, resp. remove the peg inA, with the above conditions. We define theexternal external heightH(A, I, J)to be the maximum 0 number of moves such that, starting fromI, the position reachedIcontains 0 Aand that there exists a legal way of derivingJfromIthus look at. We keywords: Pegsolitaire, Hi-Q, Pagoda function. AMS classification:primary 05A99, secondary 91A46, 52B12, 90C08. 1