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 Another twin puzzle variation 
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Posted on: Fri Nov 03, 2017 5:45 am




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Joined: Sun Jan 31, 2016 7:52 am
Post Another twin puzzle variation
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Another variation of a twin puzzle. This is similar to a standard twin puzzle, but the solution to each puzzle is different. Each number in the puzzle on the left is swapped for another number in the puzzle on the right, in the example puzzle (yellow background) all the cells that are 1 in the left puzzle are a 4 in the right side puzzle, all the cells that are 2 on the left are 1 on the right, 3 left = 5 right, 4 left= 3 right, 5 left = 2 right.

I would be interested to know what people think of this puzzle.


Last edited by michaele on Fri Nov 03, 2017 2:33 pm, edited 1 time in total.



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Posted on: Fri Nov 03, 2017 7:34 am




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Location: Ladysmith, BC, Canada
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Post Re: Another twin puzzle variation
If I understand this correctly, 2 different solutions for a pair of twin puzzles which leads me to the conclusion that we have 2 separate puzzles with 2 different solutions and therefore they really aren't twins but 2 puzzles entirely on their own.....or am I missing something here??


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Posted on: Fri Nov 03, 2017 8:10 am




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Joined: Sun Jan 31, 2016 7:52 am
Post Re: Another twin puzzle variation
I have not explained it very well. Although they both have different solutions, they are connected in that each number in the left puzzle is swapped for a different number in the right puzzle. If you look at the 5x5 example, every cell in the left puzzle that contains a 1 has a 4 in the corresponding cell in the right puzzle, and every cell with a 2 in the left is a 1 in the right etc. I was thinking that this would make one more thing to be considering when solving the puzzle, and make it a bit more interesting than just having both puzzles with the same solution. Trying to figure out which numbers are swapped with which will make you think about how you solve the puzzles a little different, for example in the 6x6 puzzle above it is reasonably easy if you consider the number swapping possibilities in mind while solving it.

I hope this helps, my explanation is a little vague. Let me know if more explanation would be helpful.


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Posted on: Fri Nov 03, 2017 11:35 am




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Joined: Thu May 12, 2011 11:58 pm
Post Re: Another twin puzzle variation
michaele wrote:
I hope this helps, my explanation is a little vague. Let me know if more explanation would be helpful.

Maybe draw a little diagram for the number swaps of the sample puzzle?

So

1 -> 4
2 -> 1
3 -> 5
4 -> 3
5 -> 2

?


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Posted on: Fri Nov 03, 2017 11:49 am




Posts: 34
Joined: Sun Jan 31, 2016 7:52 am
Post Re: Another twin puzzle variation
That is a good idea, I will try adding something like that to the puzzle.


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Posted on: Fri Nov 03, 2017 2:36 pm




Posts: 34
Joined: Sun Jan 31, 2016 7:52 am
Post Re: Another twin puzzle variation
I have updated this puzzle and the example puzzle to include a table of number swaps, hopefully this is an improvement.

Thank you pnm for the suggestion.


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Posted on: Tue Nov 07, 2017 11:49 pm




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Post Re: Another twin puzzle variation
Michaele,
Can I assume that no number will be mapped / swapped onto itself?


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Posted on: Wed Nov 08, 2017 12:01 am




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Joined: Sun Jan 31, 2016 7:52 am
Post Re: Another twin puzzle variation
Yes, that is right. All numbers will be changed to a different number. That might seem a minor point, but at times it will be very useful to solving the puzzles.

I will post some more of this type of puzzle later this week.


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Posted on: Sun Jun 03, 2018 10:10 pm




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Post 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.
[confused]
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”.
[huh]

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” [biggrin]

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Posted on: Mon Jun 04, 2018 10:32 am




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Joined: Sun Jan 31, 2016 7:52 am
Post Re: Another twin puzzle variation
Thanks for your feedback about my puzzles. I was concerned that my description of this type of puzzle was not very clear, and that some people did not try the puzzles because it was confusing.

If I create more of this type of puzzle I will try to come up with a better name, and a better way of describing the concept.

If anybody is interesting in seeing more of this type of puzzle then please send me a message and I will create some more.


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