Heightmazes

Notes on possible new logic puzzles

Heightmazes

Postby dyitto » Sat May 22, 2010 9:54 am

(Recovered after shutdown of the original players forum)

Suppose we have a grid filled with numbers, and let's say that the numbers indicate heights of buildings:

Image

Suppose we could jump from one building to an adjacent one (horizontally or vertically) if and only if the difference between the two heights equals 1 or zero.
Thus we could draw lines indicating where we can and cannot jump:

Image

The resulting maze with numbers could be called a heightmaze.

The following two puzzles deal with the opposite activity: we know the lines of the maze and we must fill in the missing numbers.

Image

Text format of the same puzzle:
Code: Select all
26 .. .. ..|31|36 36|33 ..
--    --       --       --
.. ..|.. ..|31|.. ..|.. ..
   -- --       --         
22|30 .. .. ..|.. ..|..|..
-- -- --                 
.. ..|.. ..|30|38|.. ..|33
      --       --    -- --
24|27|.. ..|..|.. ..|30|27
--    --          --     
30|27 27 ..|32 .. .. 30|28
      -- -- -- --    --   
.. ..|34 33|29 .. ..|..|28
--    --    -- --         
31|..|..|..|..|..|.. ..|..
                         
31 ..|35 35 .. .. ..|.. ..


Image

Text format of the same puzzle:
Code: Select all
32|..|..|35|..|.. ..|.. 47
                  --    --
.. .. .. ..|.. .. ..|.. ..
--    --          -- --   
.. ..|.. ..|..|.. ..|.. ..
-- -- --    --       --   
.. .. .. .. .. ..|.. .. ..
      -- --    -- -- -- --
..|.. .. ..|36 .. .. .. 33
--    --    --    --    --
..|.. ..|34|..|..|31 .. ..
      -- --       -- --   
.. .. ..|41|.. ..|41 41|32
   --       --    --    --
..|35 ..|.. .. .. .. .. 40
   --    --               
30 30|34|.. ..|..|..|..|..


(Regards, evert)
Last edited by dyitto on Tue Aug 14, 2012 9:04 pm, edited 1 time in total.
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Re: Heightmazes

Postby RW » Wed May 26, 2010 6:41 am

Both of your puzzles have hundreds, if not thousands, of solutions. I think this puzzle would need some extra constraints to become interesting. At the moment it would be far too easy to complete a puzzle with an unique solution (and it is way too easy to complete these multisolution puzzles as well).

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Re: Heightmazes

Postby dyitto » Wed May 26, 2010 5:27 pm

RW wrote:Both of your puzzles have hundreds, if not thousands, of solutions.RW

Hi RW,
Could you post at least two different solutions for one of these puzzles?
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Re: Heightmazes

Postby RW » Tue Jun 15, 2010 5:30 pm

dyitto wrote:Hi RW,
Could you post at least two different solutions for one of these puzzles?

Ah, yes, there is of course the constraint that you must not be able to jump over the lines. How silly of me not to notice that... I thought it seemed rather suspicious that someone posted a puzzle like this and should have suspected that I'm missing something very obvious... :oops:

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Re: Heightmazes

Postby evert » Sun Jan 20, 2013 10:11 am

Another heighmaze built with Lego:

Code: Select all
08|10 09|12|10 10 10|07
               --     
09 10|08|12 11|08 09 08
--    --    --       --
01|11 12 13 14|07|10 11
   --    --    --    --
01|04|11|16 15|12 11|15
      -- --       --   
02 03 04|07|14|12 13 14
   --       --    -- --
02|06 05 06|09|11 10 11
--    -- --    --    --
06 07 08 08 08 08 09 08
            -- --    --
05|07|09|07 06|11 10 11

Image

Image
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Re: Heightmazes

Postby Smythe Dakota » Sun Jan 27, 2013 11:19 am

Dytto:

I love the concept! I'm always looking for new logic puzzles.

The first one took me a little while. I can tell you, for sure, for anybody still wondering, there is only one solution.

The second one has me stumped, so far. I have solved roughly the lower right quarter, and about half the lower left quarter. There are a couple of places (the 36 in the center through the 47 in the upper right corner, and the 30 in the lower left corner through the 35 near it) where the slack is only 1, i.e. the path must be strictly monotone increasing except for one point somewhere along the path. I'll keeping working on it.

The third one (with the Lego photo) I haven't tried yet. I've printed it out and will look at it between rounds of the chess tournament I'll be playing in today.

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Re: Heightmazes

Postby evert » Mon Jan 28, 2013 7:11 pm

The lego one is not a puzzle but a completed one ;)
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Re: Heightmazes

Postby Smythe Dakota » Tue Jan 29, 2013 1:35 am

Oops, I obviously didn't look at it hard enough. :oops:

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Re: Heightmazes

Postby evert » Wed Feb 20, 2013 10:19 pm

Smythe Dakota wrote:The second one has me stumped, so far. I have solved roughly the lower right quarter, and about half the lower left quarter. There are a couple of places (the 36 in the center through the 47 in the upper right corner, and the 30 in the lower left corner through the 35 near it) where the slack is only 1, i.e. the path must be strictly monotone increasing except for one point somewhere along the path. I'll keeping working on it.

I must admit that the second one contains an inelegant chain, however still doable with pen & paper.
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Re: Heightmazes

Postby civiliza » Thu Feb 21, 2013 8:31 pm

evert wrote:I must admit that the second one contains an inelegant chain, however still doable with pen & paper.


Managed to do around 2/3 of it with logic, then resorted to experimentally plugging in numbers. Due to this, I have got an answer, but couldn't guarentee that it is the only answer.
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Re: Heightmazes

Postby evert » Sat Feb 23, 2013 9:24 am

Two more easy puzzles - definitely no trial & error:

Puzzle 3:
Code: Select all
00 09|14|22 22 00 00 00 22
   --    -- -- --       --
00 00 00|00 00 00 00|00 00
   --       -- -- --     
11|15 00 00|00 00 09|22|20
-- --    --          -- --
00 09|00 13|00|08|00|00|01
   --          -- --     
00 00 00|00|00 00 00 00 00
-- --    -- -- -- --     
13 00 12 00 00 14|10|00|00
      --    -- --    --   
00|00 00|00 00 00 09|04 00
--    -- -- --    -- --   
15 00 00|14|07 00 00 00 00
--          -- --    --   
00 14|00 00 15 15|07|00 05

Image

Puzzle 4:
Code: Select all
21 00|00 00|20 20|14 00|11
--    --       --         
00 00 00 00 00|00 00|00 00
--       -- -- --    --   
20 20|00|00 00 00 13|09 00
   --             -- --   
21|16 00 00|00|00 00|03|09
-- --       --           
14 00 00|00 18|00|11|03|10
   -- --    -- --       --
00 00|18|00 15 00|11|00 02
--          --          --
18 00 00|00|00 00|00|00 00
--    -- --    -- -- --   
15|00 00 17|00 00 00|00 00
      --    --       --   
15 00|00 17|00 11|00 00 00

Image
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Re: Heightmazes

Postby Smythe Dakota » Sun Feb 24, 2013 3:55 am

OK, I solved #3 and #4. #3 was fairly easy, #4 was just about right -- challenging enough to be interesting, but easy enough not to be tedious.

In both of them, however, I noticed that, had I wanted to, I could have used the much-deprecated (in both Sudoku and Kakuro) "uniqueness technique" to immediately place an X in cell A, because a Y in cell A would permit either of two values in cell B, without it ever being possible to eliminate either.

The purist in most of us obligates us not only to find a solution, but to demonstrate that it is unique. Assuming uniqueness, then, becomes a big no-no.

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Re: Heightmazes

Postby civiliza » Sun Feb 24, 2013 8:05 pm

re Uniqueness:

Hmm, must have missed that. At one point I thought that the 2 to 11 run in puzzle #4 had more than one solution, but later discovered that only one solution would work with the rest of the puzzle. Mind you, I was just glad to complete the puzzle, so I wasn't really sensitised to uniqueness issues.
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Re: Heightmazes

Postby evert » Sat May 18, 2013 10:40 pm

Another 10x10 puzzle:

Image

The solution is presented here in lego:

Image

Image
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Re: Heightmazes

Postby ixsetf » Sun May 18, 2014 9:36 pm

Maybe I am confused about the rules, but for squares like r1c5 wouldn't there be multiple valid answers? Neither a 1 or a 2 would produce an immediate contradiction, and since it is not connected to the rest of the grid its value has no effect on the height values on the rest of the puzzle. I am unclear how you would go about making a determination about which value is correct.
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