- Moving pairs

Given something like a checkers board, moving pairs would be checkers
paired together and arranged on the board

Given something like a checkers board, moving pairs are checker pieces
said to be paired.
The pairs don't have to be next to eachother.

Any way arranged is fair for how this works, but it matters for how they work.

There's no such thing as an empty space.

They are the idea of how they move, and the problem with finding how
to move them.

- Moving a pair

Find the pair to be what doesn't have any way to move.

Each of the pair is to move together at the same time.

A pair can only move to another pair.

A pair moves to another pair, and each of the pair becomes paired with
each of the other pair. So now both pairs are new pairs, with a
pair that moved to a pair not being a pair anymore, but both a pair
with the pair that moved to another pair.

A pair moves to another pair, but the other pair is what moves away at
the same time.

A pair moves to another pair, the pair it moves to has to move at the
same time to another pair, and left to move as a pair
but became the new pair with the pair that goes where it's leaving from.

You can't know what any pair's first move is until you know it's last move.

There's nowhere to think in the way moving pairs can move how it has
any inbetween to stop moving. It's always that a pair moving is making
another pair move, and is having a pair move it.

For any pair there's always one way to move it.

To think of pairs in the middle of moving is to think of needing to
know the end and beginning at the same time.

When a pair moves it's what is moving away from what had to move it,
and is moving what needs to move at the same time for whre it's going.

Each time a pair is moved, all the pairs involved in moving are
alternated as new pairs.

You can't know how a pair moves, it's to figure out as the problem
they have. The last pair to move has to be discovered before the first
pair to move has a place to move.

The answer to how to make a pair move is to find how to move the pair,
that goes to another pair and becomes a new pair, and carries on until
a pair goes back where the first pair left. That's to even know the
first place a pair can move.

All the pairs have a way to move, but may involve more or less of the
other pairs to move at the same time.

- Moving pairs are a machine

They can be arranged in any way so that the movement of a few pairs
makes for the movement of a few other in any way.

Pairs can store how to associate any letter with any number by having
it so some pairs say a letter and other pairs say a number, so how you
move some pairs to say which letter is for some other pairs to move to
say which number.

Any pair that is moved can make it so other pairs to move have another
answer to how they move.

Pairs that work in moving together can be close or far apart, so they
extend across the idea of how it's a machine as much as can make sense
for any way to have a machine work.

- Machine diagram

See each pair for how it can be moved with the other pairs that move together.

Draw pairs that move together with a line, or say one piece moves to
another piece.

Say for each group of pairs that move together.

- The trick

See the machine diagram? Make a pair move then see the machine diagram again.

How did the machine move?

Enjoy

Given your tenacity at posting this about the web I thought I would take a stab at formalizing it for you. In order to represent your "machine" in code you have to distill it down to a formal system which can be expressed within a set of rules.

So lets begin by what you mean by a board:

My assumption: A board (B ) is a set of ordered pairs from Z x Z.

so for example B may be {(1,1), (1,2), (1,3), (2,1), (2,2), (2,3)}.

Now to define a Pair the definition is not clear, it is clear that a Pair contains at least two points from the board say say (1,1) and (2,3). However this alone does not seems to be enough.

Lets define a set T called tokens {a, b, c, d, e, ... etc}.

Let a Pair be a triplet T x B x B: so a member of the set of all Pairs (P) may look like: (a, (1,1), (2,3))

Now there are something that are not clear.

Q1: for each token t in T, does there exist a unique Pair p in P?

that is, can I have (a, (1,1), (2,3)) AND (a, (1,2), (2,1)) both in P or not?

Q2: for each point (x, y) in B does there exist a unique Pair in P such that it contains (x, y)?

From what I can gather from your discussion the answers should be "yes" to Q1 and Q2. -- each token has a unique Pair, and each point in B has a unique Pair.

For all points b in B there exists a Pair p such that p is in b. -- i.e. there are no empty squares in B.

We will also assume that for all t in T there exists a Pair p such that t is in p.

So we have trivially that the cardinality of T equals the cardinality of P and the cardinality of P is twice that of B (since there are 2 board points to one Pair). This is always true (therefore there is a conservation of pairs The number of pairs never changes).

So if I want to move a Pair p I need to find another Pair q and replace the token in q with the token in p, BUT the token in q must go somewhere.

here is where you description falls apart formally. You seem to go into great detail describing the moving process, but I don't see any reason given why I can't just swap the tokens of p and q or choose some subset of B containing p and q and randomly choosing a permutation of the tokens.

Can you use the above description of the Board, Tokens, and Pairs to describe Pair motion?

I am assuming there are some rules but you don't do a good job of explaining them so that I can formalize them.

This post has been edited by NickDMax: 23 Jul, 2008 - 10:35 PM