## Heightmazes

Notes on possible new logic puzzles

### Heightmazes

(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:

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:

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.

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 .. .. ..|.. ..`

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.
evert on the crashed forum

dyitto

Posts: 118
Joined: 22 May 2010
Location: Amsterdam

### Re: Heightmazes

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).

RW
RW
2010 Supporter

Posts: 1010
Joined: 16 March 2006

### Re: Heightmazes

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?
evert on the crashed forum

dyitto

Posts: 118
Joined: 22 May 2010
Location: Amsterdam

### Re: Heightmazes

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...

RW
RW
2010 Supporter

Posts: 1010
Joined: 16 March 2006

### Re: Heightmazes

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`

evert

Posts: 186
Joined: 26 August 2005

### Re: Heightmazes

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.

Bill Smythe
Smythe Dakota

Posts: 564
Joined: 11 February 2006

### Re: Heightmazes

The lego one is not a puzzle but a completed one
evert

Posts: 186
Joined: 26 August 2005

### Re: Heightmazes

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

Bill Smythe
Smythe Dakota

Posts: 564
Joined: 11 February 2006

### Re: Heightmazes

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.
evert

Posts: 186
Joined: 26 August 2005

### Re: Heightmazes

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.
civiliza

Posts: 64
Joined: 25 October 2010

### Re: Heightmazes

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`

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`

evert

Posts: 186
Joined: 26 August 2005

### Re: Heightmazes

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.

Bill Smythe
Smythe Dakota

Posts: 564
Joined: 11 February 2006

### Re: Heightmazes

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.
civiliza

Posts: 64
Joined: 25 October 2010

### Re: Heightmazes

Another 10x10 puzzle:

The solution is presented here in lego:

evert

Posts: 186
Joined: 26 August 2005

### Re: Heightmazes

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.
ixsetf

Posts: 50
Joined: 11 May 2014

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