Chain Puzzles

wintersolstice · 2009-06-16 21:44:44

There's a type of puzzle that I call a "chain puzzle". Basically you have a given number of chains and your given how many links they have. Your job is work out, "What is the minimum number of links that need to be broken and rejoined to make a continuous loop?" The notation I'd use for stating a puzzle is to list the number of links in each chain.

So 3,4,5 would mean there are 3 chains, one has 3 links, one has 4 and the other has 5.

I would say there are three levels of difficulty:

"all the same" = easy
"From one to a given number consecutive" = medium
"totally random" = hard

easy: 4,4,4,4,4

medium: 1,2,3,4...99,100 (all the numbers from 1-100)

Hard: 4,5,6,7,12,16,21,41,61,63,71,72,71,72,73,74,80,81,83,85,90,100,101,109,121,131,132, 133,135,136,138,140,144,145,150,161,162 (you can take your time with this one)

Tell me what you think:D

wintersolstice · 2009-08-03 08:10:14

I posted the answer to the first in "Soroban"'s thread and I give you the answer to the second: 88
just break up 12 chains that add together to make 88 and put them between the remaining 88 chains!

So see if you can solve the 3rd

Ask if you want to to give some advice on solving them.

bobbym · 2009-08-03 13:20:38

Hi wintersolstice;

Sorry but I am not following you on the 88 answer. How did you get it?

phrontister · 2009-08-03 14:57:56

Hi wintersolstice,

Here's how I tackled a shorter example of a medium-level problem:-

Problem: 1,2,3,4,5,6,7,8,9,10

Solution: 7 (chains 1,2,4)...ooo1ooooo2oooooo2ooooooo4oooooooo4ooooooooo4oooooooooo4

I've only just started to have a look at your puzzle and haven't got any further than this with it.

Btw, a 1-linked 'chain' isn't really a chain - but that's just being picky.

phrontister · 2009-08-03 16:46:20

Hi wintersolstice,

I think I've worked out a formula for solving medium-level problems that commence with "1":

Last edited by phrontister (2009-08-04 14:16:20)

bobbym · 2009-08-03 19:42:50

Hi phrontister;

Very nice. If winter okays it then I have to say well done.

wintersolstice · 2009-08-04 04:33:45

phrontister wrote:

Hi wintersolstice,
I think I've worked out a formula for solving medium-level problems that commence with "1":

That's absolutly correct:D Although my method was slightly different, but I still used triangle numbers.

When I can I'l post a higher dimensional analog.:D

phrontister · 2009-08-04 06:32:58

wintersolstice wrote:

When I can I'l post a higher dimensional analog.:D

I'm sorry, but I don't know what that means (my maths knowledge is only Year 4 high school level ). You'd have to keep it pretty simple for me to understand it, or include some explanations of any advanced concepts.

I didn't know how to prove my formula but was interested to see if it worked for all cases. So I manually tested the first 30 groups...and found a flaw if L was 1 less than a triangular number. However, I worked out how to fix that and have amended my formula in the other post. I also tested 4 groups within the 60 to 80 range...successfully.

Only groups {1,2}, {1,2,3} and {1,2,3,4} now fail, but I guess that those numbers are too low for the formula to handle.

The new formula is a bit simpler, which is nice.

Last edited by phrontister (2009-08-04 06:43:45)

wintersolstice · 2009-08-04 07:01:49

phrontister wrote:

wintersolstice wrote:
When I can I'l post a higher dimensional analog.:D
I'm sorry, but I don't know what that means (my maths knowledge is only Year 4 high school level ). You'd have to keep it pretty simple for me to understand it, or include some explanations of any advanced concepts.

well I was going to post a very simple higher dimensional puzzle anyway, and to help people get the hang of it I'll give an example of a solved puzzle:D

OH and I'll post a picture to go with it

Last edited by wintersolstice (2009-08-04 07:03:26)

phrontister · 2009-08-04 07:13:05

Here's an image of the Excel worksheet that I used for testing my formula on the first 30 groups.

Last edited by phrontister (2009-08-04 13:49:55)

wintersolstice · 2009-08-09 22:34:54

Here's a higher dimentional analog (if the original puzzle could be regarded as 2D, this would be 3D) When I can figure it out I'll see if I can do 4D!!!

If it doesn't make sense, you have 42 rectangles (of chain links) 3x5 and you have to build a torus which contains 18 rings each containing 35 links.

this is one of the easiest of this level, plus I came up with the idea of higher dimensional analogs myself.

Hope this might be more enjoyable

Please post if you have any questions

Edit: I've just spotted a mistake, (I didn't try and solve it first!) change the number of rectangles from 42 to 144.

OH yeah and change the torus to 36x60 (not 18x35)

Last edited by wintersolstice (2009-08-09 22:53:50)

Math Is Fun Forum

#1 2009-06-16 21:44:44

Chain Puzzles

#2 2009-08-03 08:10:14

Re: Chain Puzzles

#3 2009-08-03 13:20:38

Re: Chain Puzzles

#4 2009-08-03 14:57:56

Re: Chain Puzzles

#5 2009-08-03 16:46:20

Re: Chain Puzzles

#6 2009-08-03 19:42:50

Re: Chain Puzzles

#7 2009-08-04 04:33:45

Re: Chain Puzzles

#8 2009-08-04 06:32:58

Re: Chain Puzzles

#9 2009-08-04 07:01:49

Re: Chain Puzzles

#10 2009-08-04 07:13:05

Re: Chain Puzzles

#11 2009-08-09 22:34:54

Re: Chain Puzzles

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