Re: Another twin puzzle variation

I love the idea, but the term “swap” is somewhat confusing.

That's how my thoughts were running:

Ambidexter wrote:

A

**swap** is a procedure where two variables

*exchange* values.

With reference to the puzzle, this would mean that, say, cell A4 of left puzzle contains “3”, cell A4 of right puzzle contains “5”, and then we

*swap* them, i.e. now “5” is in the left grid and “3” is in the right grid.

OK, maybe we swap two elements of the central diagram, not the grid?.. In terms of group theory, it may be interpreted as a

**transposition**.

However, the process (3 ↔ 5) is symmetrical, i.e. if left “3” becomes right “5”, then left “5” must turn into right “3”.

What if we have a broader look at the diagram?

A

**permutation** is a

bijective correspondence where elements of a finite set get shuffled in any way (including staying in their own place), forming cycles (loops).

Example for set {1,2,3,4,5,6}: 1 → 5, 5 → 3, 3 → 1, 2 → 6, 6 → 2, 4 → 4 (actually, the parts should be sorted in ascending order), or, in cycle notation, (153)(26)(4).

A

**derangement** is a permutation with no fixed points (i.e. no element may stay in its place).

So this diagram is in fact a derangement. Not a swap, not even a list of swaps.

If you don't like the sound of the word, just call it “permutation”. Though “deranged twin” sounds cool

Moreover, if one puzzle uses numbers {1…7} and its twin uses {−3…3}, i.e. elements of one set get replaced with elements of a

*different* set, I'd call it

**substitution**.

So, when you are saying…

michaele wrote:

each number in the left puzzle is swapped for a different number in the right puzzle

…I guess it should go like this:

Quote:

each number in the left puzzle is *substituted* with a different number in the right puzzle

Correct me if I'm wrong (after all, I'm not a native speaker).

I'd also suggest searching for terms related to “

fixed points”.

Or leave aside all this nerdy Maths stuff and christen them “non-identical twins”