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Infinite loops

Posted By: playBunny
Date: Sunday, 3 March 2013, at 4:55 p.m.

In Response To: Infinite loops (Bob Koca)

Let a node be [position, player on roll, roll]. Examples A, B and C as the respective positions in the loop that I posted.

An atomic infinite loop is a sequence of nodes such that each node is unique and there's a legal move for each node which obtains its successor. Examples L1 = {A, B, C, A}, L2 = {A, B, D, E, F, A}, where the final A represents the closure rather than a duplication.

A compound infinite loop is one consisting of a sequence of two or more atomic loops which have a node in common. Example, {L1, L2} = {A, B, C, A, B, D, E, F, A}

There are an infinite number of such compound loops and sequences of atoms are a countable set therefore the set of infinite loops is a countable infinity.

All that remains is finding a node from which two atomic loops are possible, which is trivial. For example, in the loop that I posted, the final move C = {Blue, 65: 18/7*} can be replaced by D = {Blue, 31: 18/14}, E = {White, 21: 18/11}, F = {Blue, 34: 14/11*/7}.

(The wider definition of a compound loop is the same as an atomic loop except that there is duplication of one or more nodes, for example {P, Q, R, S, Q, T, P} where {Q, R, S, Q} is an atomic loop but P and T are independent nodes, however I think the discussion above is clearer with the simpler definition)

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