Fairy chess piece tour puzzles including Numbrix & Hidato

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Fairy chess piece tour puzzles including Numbrix & Hidato

Postby tarek » Mon Jan 09, 2012 9:22 pm

Wikipedia wrote:The knight's tour is a mathematical problem involving a knight on a chessboard. The knight is placed on the empty board and, moving according to the rules of chess, must visit each square exactly once. A knight's tour is called a closed tour if the knight ends on a square attacking the square from which it began (so that it may tour the board again immediately with the same path). Otherwise the tour is open. Variations of the knight's tour problem involve chessboards of different sizes than the usual 8 × 8, as well as irregular (non-rectangular) boards.

The variations can also be extended to include other well known chess & Fairy chess pieces.

In this thread there will be puzzles based on these tours which can be solved by knowing some of the visited squares on the tour.

There should be some knowledge about the chess piece movement in order to solve the puzzles.

Here are some of the terms you may encounter:

nxn: size of the chessboard, a regular chessboard is 8x8

Tour: all squares on the board must be visited once starting at square 1. the last visited square has an ordinal equal to the total number of squares on the board.

Closed: The 1st square Must be one move away from the last square

Open: The 1st square can be of any distance away from the last square

Obstacle: Each visited square will form an obstacle in the path of future moves

Wazir: A Fairy chess piece that moves exactly 1 square in an orthogonal direction

Hidato: An open king's tour puzzle where the 1st & last sqaures are known at the start.

Numbrix: An open Wazir's tour

Toroidal: The Board top/bottom & Rt/Left edges are attached this wraps the board to make a toroid (doughnut/halo like) structure.

Links:

http://forum.enjoysudoku.com/hidato-t6404.html : Here evert (dyitto) & udosuk discussed several solving techniques for Hidato which can be used in other variations.

http://www.dyitto.com/downloads/hidatordownload/ : dyitto's excellent solver generator, covers open Wazir/King puzzles with several solving techniques

http://forum.enjoysudoku.com/queen-s-tour-hidato-variant-t30307.html : Here dyitto demonstrates a Queen's tour with obstacles puzzle

The following is how the original thread started.

I have been made aware of the Numbrix puzzle which is an open Wazir's tour (orthogonal 1 square move) on a 9x9 grid

The Hidato puzzle is a King's tour (Any neighbouring 1 square move) on a variety of grids.

These 8x8 (Chess board) puzzles have been verified to be minimal and to have a unique solution by my solver, I couldn't find an independent solver to verify this. :(

here is a sample open King's tour puzzle & solution
Code: Select all
22 21 00 00 27 00 63 00
00 00 00 01 00 00 00 00
38 00 00 00 00 00 00 00
00 00 00 11 00 00 00 00
49 00 00 00 00 00 00 00
00 00 00 00 00 13 58 00
45 00 00 00 00 33 00 00
00 00 00 00 00 00 00 00

22 21 25 26 27 03 63 64
23 24 20 01 02 28 04 62
38 09 10 19 18 05 29 61
39 37 08 11 06 17 30 60
49 40 36 07 12 31 16 59
48 50 41 35 32 13 58 15
45 47 51 42 34 33 14 57
46 44 43 52 53 54 55 56


Here are some open King's tour puzzle puzzles of untested difficulty
Code: Select all
18 00 00 00 00 00 00 00
00 00 00 00 01 00 00 00
00 00 00 00 40 00 00 00
47 00 00 00 00 00 00 00
50 00 00 00 00 00 07 00
00 00 00 00 00 00 11 00
00 00 00 00 00 00 12 00
57 00 00 00 34 00 00 30

00 00 61 00 00 00 00 27
39 00 00 00 00 00 28 00
00 00 00 00 00 07 04 00
00 00 00 00 00 00 00 00
00 00 00 00 14 33 00 09
49 00 00 00 00 00 00 00
00 00 00 00 12 00 00 00
00 00 00 00 45 00 00 00

00 00 30 00 00 00 00 64
00 00 00 04 00 00 00 00
00 00 00 00 00 00 07 13
01 00 26 00 00 00 08 00
00 00 00 00 00 00 00 00
00 00 37 00 00 00 10 00
42 00 00 00 00 00 00 21
00 00 00 39 00 00 00 00

63 00 00 22 00 06 00 30
00 00 09 00 00 00 00 00
00 00 00 00 00 00 00 00
11 00 00 01 44 00 00 00
00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00
52 00 00 00 00 38 00 00
00 00 00 00 15 00 00 00

55 00 00 20 00 00 00 39
00 17 00 00 00 00 00 24
00 00 00 01 00 00 00 00
00 00 00 00 00 00 26 00
50 00 00 00 00 00 00 00
00 13 00 00 00 00 00 06
00 00 00 00 00 00 00 64
00 00 00 00 00 00 00 00

00 00 00 00 51 00 00 00
00 00 00 00 00 00 08 00
14 00 00 00 00 00 00 24
00 40 00 00 10 00 00 00
00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00
00 00 00 00 00 00 02 00
00 00 19 00 00 01 00 34

00 18 09 10 00 00 13 28
00 00 00 00 11 12 00 00
00 00 00 00 00 00 01 00
00 00 00 00 00 00 00 49
00 00 00 00 00 00 00 00
00 00 00 44 04 51 00 00
00 00 00 00 00 00 00 00
64 00 00 42 00 00 00 00

00 22 00 00 00 16 30 00
00 00 00 00 10 00 28 00
01 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00
00 00 03 00 37 00 00 00
43 00 00 00 49 00 00 52
00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00

00 00 19 00 26 00 00 63
00 00 00 00 00 07 05 00
00 00 00 00 00 00 02 00
43 00 00 00 00 00 34 00
00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00
00 00 00 00 39 52 00 00
00 00 00 00 00 00 00 00

29 00 00 44 00 00 00 00
00 00 00 00 00 00 00 00
61 00 00 00 00 00 00 00
00 00 00 00 00 49 00 00
00 00 00 03 00 00 00 00
00 00 00 13 00 00 00 00
00 00 00 00 21 00 36 00
00 00 06 07 00 00 00 00

00 00 00 00 00 12 00 00
00 00 00 00 00 00 00 00
00 00 07 00 00 00 54 00
00 03 00 00 00 00 38 00
00 00 00 35 25 00 00 00
17 00 00 00 00 00 00 46
32 00 00 00 00 00 00 00
00 00 28 00 00 00 00 00

22 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00
57 00 00 00 00 00 00 00
00 00 48 06 00 00 00 00
40 00 00 00 00 17 00 29
00 00 51 00 00 00 01 00
00 00 00 00 00 00 00 00

63 00 19 00 00 00 00 09
00 00 00 21 00 03 00 00
00 00 00 00 00 40 07 00
00 00 00 58 00 00 00 00
00 28 00 00 00 00 00 00
00 00 00 46 00 00 00 00
00 31 00 00 00 00 00 00
00 00 00 00 00 00 00 00

00 00 00 51 00 00 27 00
00 00 00 00 00 00 12 00
00 00 00 00 00 00 00 00
00 00 43 00 00 00 03 00
41 00 00 00 00 00 00 17
40 00 00 00 34 00 00 08
00 00 00 00 00 00 00 64
00 00 00 00 20 00 00 00

63 00 00 00 08 00 56 00
00 00 00 00 00 15 00 00
00 00 12 00 00 00 00 00
00 01 02 00 00 00 00 00
00 00 00 30 00 00 00 34
00 00 00 00 00 00 00 00
00 00 00 22 41 00 00 37
25 00 00 00 00 00 00 00

00 00 00 22 00 48 00 00
00 63 00 00 00 00 00 00
58 00 00 00 00 00 00 00
59 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00
00 00 00 00 00 00 33 00
00 09 00 06 00 00 29 00
41 00 00 00 00 14 31 00

00 53 00 00 44 45 26 00
00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00
50 00 00 00 30 00 05 00
00 00 00 00 00 15 00 00
00 00 00 00 00 00 00 03
00 00 00 00 00 00 00 17
64 00 00 00 35 01 00 00

19 00 00 00 36 00 42 00
00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 39
00 00 00 00 00 00 00 00
00 00 00 00 14 00 00 00
00 00 00 00 00 26 12 00
53 00 00 00 10 00 64 00

00 00 00 01 00 26 00 00
00 00 00 00 00 00 00 00
00 00 00 00 00 00 31 00
21 00 00 00 00 00 00 00
56 00 00 00 00 11 00 00
00 00 00 00 41 00 00 00
00 00 00 00 00 00 00 36
49 00 00 00 08 00 00 00

16 00 00 38 00 45 00 00
00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00
00 00 23 13 00 00 00 55
00 24 00 00 00 00 00 00
05 00 00 00 00 00 00 61
00 00 00 51 00 00 00 00
30 00 00 00 00 00 63 00

00 00 00 00 44 00 41 00
30 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00
00 27 00 00 00 00 00 00
00 00 00 10 00 00 00 00
20 00 00 00 00 00 00 00
02 01 00 00 07 12 00 54
00 00 00 00 06 00 00 00

19 00 00 00 00 00 00 00
00 00 00 00 00 36 00 00
00 00 00 00 00 00 00 00
57 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00
00 01 00 46 00 12 00 27
51 00 00 00 00 00 28 00
00 00 00 00 00 44 30 00

00 00 00 57 00 21 00 13
01 00 00 00 00 00 00 64
52 00 00 00 00 00 11 00
00 00 00 00 00 00 00 00
00 04 00 00 00 00 00 00
00 00 00 00 00 00 00 00
00 00 00 00 37 00 00 40
00 32 00 00 00 00 00 00

00 00 00 00 49 53 52 00
00 00 00 00 00 00 00 00
00 00 12 00 00 00 00 00
00 00 00 00 04 00 00 01
10 00 00 00 00 00 60 00
17 00 00 06 00 00 00 00
00 08 00 00 00 00 00 00
00 00 00 40 00 00 00 36

00 00 00 00 00 00 00 00
48 00 00 00 00 00 00 37
49 00 00 00 00 00 00 00
46 00 00 00 17 00 00 34
00 44 00 00 26 00 00 00
05 00 00 00 11 00 00 14
00 00 00 00 00 00 00 00
00 23 00 00 00 00 00 00

48 00 61 00 00 00 00 00
00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 39
23 00 01 00 00 00 00 00
00 00 00 00 00 00 57 00
27 00 00 00 00 00 00 00
28 00 00 00 00 00 05 00
00 00 00 00 12 00 00 15

21 00 00 03 34 00 00 00
00 06 00 00 00 00 00 00
00 00 00 00 00 00 00 00
00 00 00 00 00 00 28 00
00 00 00 00 00 00 00 14
00 00 00 00 00 50 00 00
00 00 00 00 11 00 00 00
63 00 00 00 00 00 00 00

00 00 05 00 00 56 00 00
00 00 00 00 00 00 00 00
03 18 00 00 00 00 62 00
00 00 00 39 00 00 53 00
00 09 00 00 00 00 00 00
30 00 00 00 12 00 00 00
00 00 00 00 00 00 00 00
00 00 00 00 00 00 45 00

00 26 00 00 00 43 00 00
00 00 00 00 12 00 00 00
00 00 00 00 00 00 00 00
30 00 00 00 00 00 00 00
00 06 00 00 00 00 00 15
07 36 00 00 00 00 00 00
01 00 00 00 00 00 55 57
00 00 00 52 00 00 00 00

00 00 00 00 00 00 00 00
20 00 00 00 00 33 00 00
00 00 00 00 00 02 00 00
00 00 07 00 00 00 00 00
09 08 00 00 00 00 00 40
00 00 00 00 45 43 00 00
25 00 00 00 00 00 00 00
00 00 12 00 00 00 00 00

00 00 33 00 00 21 00 63
00 00 08 00 00 00 00 00
00 00 00 00 00 00 00 00
00 00 00 00 00 00 58 00
00 00 29 27 00 11 57 00
00 00 40 00 00 00 56 00
00 00 00 00 48 01 00 00
00 00 00 00 00 00 52 00

00 00 00 16 00 00 53 57
00 00 00 00 00 00 00 00
00 00 00 04 00 00 00 00
00 00 00 00 00 00 00 00
20 00 00 00 35 00 00 00
10 00 00 38 00 00 00 44
00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00

35 00 00 00 57 00 00 17
00 00 00 00 00 00 00 00
63 00 00 00 00 00 00 00
00 06 00 00 00 00 00 14
00 00 00 00 00 00 00 00
24 00 00 10 00 00 00 00
00 00 00 00 00 00 00 46
00 00 00 00 00 00 00 00

00 45 46 00 42 00 00 22
00 00 00 00 03 00 00 21
50 00 00 00 00 00 00 00
37 00 00 00 00 00 00 00
00 00 00 00 00 12 00 06
34 00 00 00 00 00 61 00
00 00 00 00 00 00 00 00
00 29 00 00 00 09 00 00


Enjoy,

Tarek

[Edit1: The title was modified to reflect that Numbrix is an open Wair's tour]
[Edit2: Corrected a small mix-up]
[Edit3: Added relationship with Hidato]
[Edit4: Removed Extra row from one of the puzzles]
[Edit5: Added an intro]
Last edited by tarek on Sun Feb 12, 2012 11:48 am, edited 4 times in total.
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Toroidal Closed King's tour 5x5

Postby tarek » Tue Jan 10, 2012 3:05 pm

Here is a sample of a minimal Toroidal Closed King's tour 5x5 puzzle.

Closed means that the last square is a King's move away from the starting square

Toroid: 3D doughnut shaped grid.

Again I couldn't find an independent solver to verify this

Code: Select all
09 .. 11 24 ..
.. 13 .. 22 ..
.. 01 05 .. ..
.. 17 .. .. ..
.. .. .. 07 ..

09 12 11 24 23
02 13 25 22 03
14 01 05 04 21
15 17 06 19 20
16 10 18 07 08
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Closed Toroidal Wazir's tour 8x8 puzzle

Postby tarek » Wed Jan 11, 2012 2:05 am

Here is a a closed Wazir's tour puzzle on a toridal chess board ... You can see how similar it is to the Numbrix puzzles.
Wazir: Orthogonal 1 square moves
Tour: Each & Every square is visited once
Closed: Last square is one move away from the starting square
Toroid: The grid is wrapped to form a 3D doughnut like grid
Fully symmetric ... I couldn't solve this manually
Puzzle is not independently verified for uniqueness/validity

Image

Code: Select all
52 .. .. .. .. .. .. 31
.. 54 .. .. .. .. 27 ..
.. .. .. .. .. .. .. ..
.. .. .. .. .. .. .. ..
.. .. .. .. .. .. .. ..
.. .. .. .. .. .. .. ..
.. 19 .. .. .. .. 40 ..
61 .. .. .. .. .. .. 44


Solution: Show
Code: Select all
52 53 64 23 34 33 32 31
51 54 01 24 25 26 27 30
50 55 02 03 04 05 28 29
49 56 13 12 11 06 07 48
58 57 14 15 10 09 08 47
59 18 17 16 37 38 39 46
60 19 20 21 36 41 40 45
61 62 63 22 35 42 43 44


[EDIT: Image updated]
Last edited by tarek on Mon Jan 23, 2012 8:24 pm, edited 1 time in total.
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Closed Wazir tour puzzle

Postby tarek » Fri Jan 13, 2012 10:05 pm

This is a closed Wazir tour on a plain chess board. I've increased the clue number to make it easier
Wazir tour: 64 Consecutive 1 square orthogonal moves covering the entire board
Closed: 1st (Sq 1) square is one move away from the last square (Sq 64)

Image

Code: Select all
.. .. 49 .. .. 60 .. ..
.. 45 .. .. .. .. 64 ..
43 .. 51 .. .. 58 .. 02
.. .. .. .. .. .. .. ..
.. .. .. .. .. .. .. ..
36 .. 34 .. .. 15 .. 05
.. 32 .. .. .. .. 09 ..
.. .. 28 .. .. 11 .. ..
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Another Closed Wazir Tour puzzle

Postby tarek » Fri Jan 13, 2012 10:28 pm

Here is one with less clues and a more difficult path
Wazir tour: 64 Consecutive 1 square orthogonal moves covering the entire board
Closed: 1st square(Sq 1) is one move away from the last square (Sq 64)

Image

Code: Select all
27 .. .. .. .. .. .. 20
.. 31 .. .. .. .. 60 ..
.. .. 37 .. .. 58 .. ..
.. .. .. .. .. .. .. ..
.. .. .. .. .. .. .. ..
.. .. 52 .. .. 01 .. ..
.. 48 .. .. .. .. 11 ..
44 .. .. .. .. .. .. 13
Last edited by tarek on Mon Jan 23, 2012 8:27 pm, edited 1 time in total.
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Re: Fairy chess piece tour puzzles including Numbrix

Postby Smythe Dakota » Sun Jan 15, 2012 5:10 pm

The nice thing is that you don't need to tell us whether you are going from 0 to 63, or from 1 to 64, because it's the same thing. (0 is the same thing as 64.)

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Re: Fairy chess piece tour puzzles including Numbrix

Postby tarek » Sun Jan 15, 2012 7:03 pm

Smythe Dakota wrote:The nice thing is that you don't need to tell us whether you are going from 0 to 63, or from 1 to 64, because it's the same thing. (0 is the same thing as 64.)

that is correct if the 0 or the 64 were not given.

It can go further by choosing the lowest numbered clue as your starting clue because it doesn't matter how big that last number would be as long as it is one move away from that lowest numbered clue.
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5x5 Open Rook Tour with obstacles

Postby tarek » Sun Jan 15, 2012 11:52 pm

The Rook is a well known chess piece
This Open rook tour has an extra constraint: The rook can't pass over a square that has been previosly visited.
In other words ... The rook leaves behind it an obstacle which it can't pass in its future moves in the tour
In other words ... There shouldn't be 2 consecutive square moves with a lower numbered square in between.
This puzzle is on a 5x5 board to limit the number of guessing.

Image

Code: Select all
15 .. 21 25 ..
.. .. .. .. ..
.. .. 01 .. ..
.. .. 12 .. ..
07 09 .. .. ..


Here is the solution
Hidden Text: Show
15 14 21 25 22
16 19 20 24 23
17 18 01 02 03
06 13 12 05 04
07 09 11 10 08


As I mentioned before ... apart from solving the puzzles manually my self ... I couldn't find any independent solvers to verify a unique solution
Last edited by tarek on Mon Jan 23, 2012 8:28 pm, edited 1 time in total.
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Re: Rook Tour with obstacles

Postby tarek » Tue Jan 17, 2012 11:15 am

I've done some tests on the( Rook & Queen) tours with obstacles which an example above highlights. Unfortunately I've manually discovered an example with maultiple solutions. This means that for the rook/queen tour with obstacle puzzles, the puzzles will have a solution but most likely there will be others.

The other tours (wazir/King & Knight) should have a unique solution.

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Re: Fairy chess piece tour puzzles including Numbrix

Postby Smythe Dakota » Wed Jan 18, 2012 5:35 am

I've done the three Wazir's tour puzzles for which you furnished full-size diagrams. The two on the plain chessboard were ridiculously easy -- even easier than Marilyn's Numbrix puzzles in the Sunday Parade newspaper supplement. The one on the toroidal board was quite a bit harder -- about right.

The 5x5 rook's tour with obstacles is much harder. Does it have multiple solutions? If so, that would explain why it's hard -- the first inclination is to try to solve them by logic alone.

One suggestion: Use all-white squares for your diagrams. When I print it out, I can't write on the dark squares visibly, regardless of pen color.

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Re: Fairy chess piece tour puzzles including Numbrix

Postby tarek » Wed Jan 18, 2012 1:07 pm

Smythe Dakota wrote:The 5x5 rook's tour with obstacles is much harder. Does it have multiple solutions? If so, that would explain why it's hard -- the first inclination is to try to solve them by logic alone.
The Rook or Queen Tours with obstacles puzzles will most likely have multiple solutions At the moment & therefore will require Guessing to get to A Solution. I'm working on it to guarantee a single solution puzzle.

Smythe Dakota wrote:Use all-white squares for your diagrams. When I print it out, I can't write on the dark squares visibly, regardless of pen color.
ooops :oops: , I haven't thought about this.

From your sucessful solving, I'm guessing that my (No obstacle) chess tour puzzle generation was sucessful. At the moment it supports (King, Queen, Wazir, Rook & Knight). The Riders therefore where the obstacle puzzles would be suitable are the Rook & Queen.

I'll be posting some 6x6 Knight tour puzzles later today.

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Re: Fairy chess piece tour puzzles including Numbrix

Postby dyitto » Wed Jan 18, 2012 10:36 pm

tarek wrote:I've done some tests on the( Rook & Queen) tours with obstacles which an example above highlights.

So did I. ;)

My solver thinks that your 5x5 open rook's tour is unique and finds the same solution, although there's a small part of my code that I should doublecheck some other time.

Two other 5x5 rook's tours:

Code: Select all
06;..;..;05;..
..;..;..;..;22
..;09;..;24;..
..;..;..;20;..
13;16;18;..;..

..;..;12;13;..
05;09;..;..;..
25;..;..;22;01
..;..;..;21;..
17;..;16;..;..
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Re: Fairy chess piece tour puzzles including Numbrix

Postby tarek » Thu Jan 19, 2012 1:21 am

dyitto wrote:
tarek wrote:I've done some tests on the( Rook & Queen) tours with obstacles which an example above highlights.

So did I. ;)

Aha ... I knew I've read about these obstacles variants before .... I have looked over through the internet & found another page about it. I should have remebered to look closer to home :D.

I think I found the source of my solver problem & luckily your solver is around to verify these thanks.

Here are the solutions for your 5x5 Rook tours & your posted queen tour

Hidden Text: Show
06 08 07 05 04
11 10 01 21 22
12 09 25 24 23
14 15 02 20 03
13 16 18 19 17

06 08 12 13 07
05 09 11 14 10
25 24 23 22 01
04 19 20 21 03
17 18 16 15 02

04 34 05 31 13 26 07 03
09 33 35 32 27 08 02 01
10 36 14 28 06 17 50 46
30 12 37 38 19 57 53 51
11 29 23 22 39 56 60 52
25 42 43 44 62 64 59 58
24 18 41 45 61 63 54 47
16 40 15 21 20 55 49 48
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Re: Fairy chess piece tour puzzles including Numbrix

Postby Smythe Dakota » Thu Jan 19, 2012 3:54 am

tarek wrote: .... From your sucessful solving, I'm guessing that my (No obstacle) chess tour puzzle generation was sucessful. ....

I am reasonably confident that the three I solved (the 8x8 Wazir, two plain and one toroidal), where you furnished full-size diagrams, each have unique solutions, because I solved each with pure logic, not resorting to T&E.

.... ooops :oops: , I haven't thought about [ square color ] ....

I worked around it by putting a blank piece of paper on top of the printout, and copying the givens to the blank page. Just enough of the diagram showed through the blank page to guide me as to where the borders lay.

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Re: Fairy chess piece tour puzzles including Numbrix

Postby dyitto » Thu Jan 19, 2012 6:46 am

This solver does wazir and king.
The obstacle variant for rook and queen is an extension of the same program - on my laptop and still quick&dirty. :oops:
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