Numbrix

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Problem Description

Numbrix™ is a puzzle currently (September, 2008) being published in the "Ask Marilyn" column of the weekly Parade Magazine and daily in the "Fun and Games" section of the Parade Magazine website.

The puzzle board is a square array of cells, 7x7, 8x8, or 9x9 in puzzles seen  so far.  For a puzzle with N cells per side, the objective is to fill  the cells with integers from 1 to NxN contiguously with each number after the first adjoining the next lower number vertically or horizontally.   Some of the numbers are pre-filled and the user's additions must interface with those to keep the entire number chain in proper order.   

A Numbrix Puzzle

Solved!

      

Background & Techniques

 The puzzles are satisfyingly solvable, some would say "too easy" but I enjoy them more than Sudoku.  Several sample puzzles are included in the program which may be accessed with the Load button

You can solve the puzzle yourself by entering numbers in available squares (any that are not prefilled). The Check button will verify any trial solution. If you want some help, the Fill forced locations button will fill in all numbers which have only one valid location based the geometry of pre-filled values. Each click will make one pass through the board filling additional numbers if possible.

The Solve button does a depth-first search search for a solution, backtracking for each dead-end path it tries.  Solve time is displayed in seconds or milliseconds so you can observe the effect of pre-filling the forced locations on solution times if desired.

Controls used for debugging were left in place for future changes/fixes but are normally invisible. The Show debug controls checkbox will make them available.  Users may drag the Show progress control bar to any slower position to watch the search process.

Non-programmers are welcome to read on, but may want to skip to the bottom of this page to download an executable version of the program.

Notes for programmers

The solution search was the most satisfying part of this project.   FindNext is the recursive function which performs a depth first search.  Because we are progressing from low to high numbers, we we don't have to worry about looping, i.e. repeating a position that we have already visited.   The pseudo code for puzzle size N looks like this:

Findnext(fromcell, fromvalue):boolean

bulletResult=false;
bulletInsert Fromvalue into Fromcell
bulletIf Fromvalue is the last point (NxN) then set result=true and exit.
bulletFromvalue not the last point:
bullet Set Nextvalue=Fromvalue+1,
bullet For each of the 4 directions check the neighbor (Nextcell)
bulletIs  Nextcell empty and Nextvalue not predefined?
bulletYes: Result=Findnext(.Nextpoint, Nextvalue)
bulletNo: If  Nextcell already contains Nextval then this is a predefined number,
bullet Result=Findnext(Nextcell,Nextval);
bulletend  neighbor search loop
bulletif Result=False and Fromvalue was not predefined then  set Fromcell  abck to empty

Procedure SolveBtnClick  makes the initial call to FindNext passing the location, or a possible location for the first entry ( "1").   Location is a one dimensional array containing NxN  the cell coordinates for all predefined numbers or (0,0) if not predefined.  Findnext uses Location to checked for predefined numbers.  It is also the used to build a list of cell locations where 1 might belong if it is not predefined.   The solution can start at any cell that is within the lowest predefined number of cells from its location.  For example, if the lowest predefined number is 5 and is located at (1,1), then 1 must belong to any empty cell within the rectangle defined by (1,1) to (5,5).    SolveBtnClick builds a list of these cells and make the initial call with each of these possible starting points until a true result is found.

The other interesting aspect of the code was setting up the controls used to bug the search code.  It is helpful to watch the search proceed if one can watch the point where the error exists.  Unfortunately that may be several thousand moves into the search and could take hours reach.   By setting limits for the display animation and forcing a pause when a specific number is reached, the error location can be located quickly (or at least a lot faster than without the controls :>).    A trackbar controls the pause time after each changed board is displayed.  Setting the bar to its max position by passes the animation completely.   When the specified value is placed in a specified cell, a pause loop is entered allowing the programmer to set a break in code or to change the pause values before continuing.   This is accomplished by setting a "Pause" flag to true and looping with  "Sleep" and "application.processmessages" commands until Pause=false.    Clicking the displayed pause message sets Pause to false.

Addendum October 9, 2008:  A Numbrix puzzle generator was posted today.  It generates puzzles of several sizes and has a few options.   Users may choose to define the border cells as the predefined values or may click cells to define any other pattern.   Puzzles up to 10x10 cells in size may be generated.  A small change to the Numbrix program allows it to solve 10x10 puzzles also. 

 Running/Exploring the Program 

bullet Download Numbrix source
bullet Download Numbrix Generator source
bullet Download  Numbrix executable
bullet Download Numbrix Generator executable

Suggestions for Further Explorations

Done October 8, 2008: A "generator" would be a nice addition  to the program.  Simply generating a random starting point and a random direction for each successive number would usually not succeed because unreachable holes would be left and stop the generation before the puzzle was complete.  So we would need to generate random moves with the constraint that each number added must not leave unreachable cells or "one way streets".  Twisty alleyways that are two cells wide, so numbers could be added down and back would seem to make a harder puzzle than those that are continuously filled as we progress.      

 

Original Date: September 28, 2008

Modified: May 18, 2009

 

 

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