The New Sudoku Players' Forum

by **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, 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.

terms you may encounter:

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

Hexagon n: The number of hexagonal tiles that constitute the Hexagonal chess board. Currently Hexagon37, Hexagon61, Hexagon91 and Hexagon127 are supported

Tour: all spaces on the board must be visited once starting at space 1. the last visited space has an ordinal equal to the total number of spaces/tiles on the board.

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

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

Obstacle: Each visited space at the end of a move will form an obstacle in the path of future moves

Wazir: A Fairy chess piece that moves exactly 1 space to an adjacent space that shares a common side. On a rectangular board this move is always orthogonal.

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

Numbrix: An open Wazir's tour

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

Leaper: A (fairy) chess piece that moves by a single Leap (jump) from one position to another in an (m,n) vector. On a rectangular board, one of the coordinate of the vector 'start space - arrival space' must have an absolute value equal to m and the other one an absolute value equal to n. From trigonometry: The leaper's Jump distance would be equal to Sqr (m^2 + n^2). A Wazir is a leaper that jumps 1 space orthogonally, you may describe the leap as a (0,1) leap. A Ferz leaps 1 space diagonally, a (1,1) leaper while the Knight is (1,2) leaper with an L shape jump ending at a space that is (Root 5) away. You may have also combination leapers that can do a variety of jumps like the King which is a (0,1) or (1,1) leaper.

Rider: A (fairy) chess piece that moves by any number of identical jumps. A Rider's extended range of movement is in reference to a leaper's movement (see above). A Bishop is a Ferz Rider, the Rook is a Wazir Rider and the Queen is a King Rider. A rider in a single move between 2 spaces may pass through several unoccupied squares. In a rider's tour: The space where the move ends is considered as the next part of the tour and therefore has the next ordinal after the square at the start of the move.

Stepper: A chess piece that moves without jumping to its destination one space at at time in a predefined fixed sequence.
It requires an unoccupied pass-through space between start and finish. the Mao is a known chess piece that moves similar to the knight.
It moves one step orthogonally followed by one step diagonally (in that sequence only)

Hopper: A chess piece that moves to destination only by jumping over an obstacle (Hopping) to the unoccupied space immediately after it.
If there are no obstacles in the possible path then the piece can't make that move.

Orthogonally adjacent spaces share a side in common, and diagonally neighbouring spaces are connected at a corner and share no sides in common.

Diagonally neighbouring spaces share a common corner. Diagonally neighbouring hexagons share no corners in common but are connected
by a line that touches a corner of each space

Colourbound piece: This means its movement is limited to spaces of only one colour on a suitably checkered regular chessboard.
The Ferz, Al-Fil and the Bishop are examples of colour bound chess pieces.

Blocked spaces: These are pre-defined board spaces that are blocked with obstacles. Theses would prevent the movement of some chess pieces. On the other hand, Hoppers may not be able to move without them. They would appear as the symbol X but occasionally depending on the theme of the puzzle, they could be a jar, rocky shore or even an acorn

Symbols used in this thread:

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.scrybqj.com/downloads/hidatordownload/ : dyitto's excellent solver generator, covers open Wazir/King/Knight 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

https://www.chessvariants.com/: Contains a glossary of terms and chess pieces. Many of the terms are quoted from here.

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 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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]
[EDIT6: Updated broken links & added the Leaper, Rider terms]
[EDIT7: Added the symbols chart]
[EDIT8: Updated the symbols chart & the "Leaper" definition]
[EDIT9: Updated the symbols chart & added new terms]

by **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

by **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

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

Solution: Show

[EDIT: Image updated]

by **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)

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

by **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)

Code: Select all: 27 .. .. .. .. .. .. 20 .. 31 .. .. .. .. 60 .. .. .. 37 .. .. 58 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 52 .. .. 01 .. .. .. 48 .. .. .. .. 11 .. 44 .. .. .. .. .. .. 13

by **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.)

Bill Smythe

by **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.

by **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.

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

Here is the solution

Hidden Text: Show

As I mentioned before ... apart from solving the puzzles manually my self ... I couldn't find any independent solvers to verify a unique solution

by **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.

tarek

by **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.

Bill Smythe

by **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

, 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.

Tarek

by **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;..;..

by **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

.

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

by **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 , 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.

Bill Smythe

by **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:

The New Sudoku Players' Forum

Fairy chess piece tour puzzles including Numbrix & Hidato

Fairy chess piece tour puzzles including Numbrix & Hidato

Toroidal Closed King's tour 5x5

Closed Toroidal Wazir's tour 8x8 puzzle

Closed Wazir tour puzzle

Another Closed Wazir Tour puzzle

Re: Fairy chess piece tour puzzles including Numbrix

Re: Fairy chess piece tour puzzles including Numbrix

5x5 Open Rook Tour with obstacles

Re: Rook Tour with obstacles

Re: Fairy chess piece tour puzzles including Numbrix

Re: Fairy chess piece tour puzzles including Numbrix

Re: Fairy chess piece tour puzzles including Numbrix

Re: Fairy chess piece tour puzzles including Numbrix

Re: Fairy chess piece tour puzzles including Numbrix

Re: Fairy chess piece tour puzzles including Numbrix