Hexagonal Boards

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Mensa Daily Puzzlers

For over 15 years Mensa Page-A-Day calendars have provided several puzzles a year for my programming pleasure.  Coding "solvers" is most fun, but many programs also allow user solving, convenient for "fill in the blanks" type.  Below are Amazon  links to the two most recent years.

Mensa 365 Puzzlers  Calendar 2017

Mensa 365 Puzzlers Calendar 2018

(Hint: If you can wait, current year calendars are usually on sale in January.)


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Problem Description

Draw hexagonal patterns for use as game or puzzle boards using regular hexagonal tiles.

Background & Techniques

Being known to the family as a puzzle lover, I received one called "Aristotle's Number Puzzle" for Christmas this year - nothing to do with Aristotle as far as I can tell. It's a hexagon shaped board made up of of 19 hexagon shaped tiles numbered 1 through 19.  The objective is to arrange them into a "magic" hexagon in which values in every horizontal and diagonal row (15 in all) sum to 38.

Our "Brute Force" program can find the 12 variations of the solution by defining 15 equations representing sums for the 19 variables . These are all isomorphic versions of the single solution, i.e. they represent rotations and mirroring of the solution. I decided to investigate further, starting by figuring out how to draw the boards.

This program does that in various sizes and with other size and orientation parameters. If we number boards based on the number of tiles, N, on one outside edge, then the total number of tiles in the board is 3N^2 - 3N +1 and the length of the longest horizontal or diagonal row is 2N+1. Search "Centered Hexagonal Number" and/or "Magic Hexagons" for more information.

Here's a puzzle board with solution hints  if you want to work on it offline.

The program has about 200 lines of code, making it of intermediate difficulty. The key problems to be solved included :

bulletScaling individual hexagon tiles with specified  size and orientation. Procedure MakeHexagon creates a TPolygon type as a 6 entry array of points representing the corners of the hexagon with given center point, radius, and angle of one vertex from horizontal.  The other 5 corners are each increased 60 degrees from the  previous corner.          
bulletDrawing tiles in appropriate position positions.  For the board with a long horizontal row. the number of rows is the same as the number of tile in the longest row.   We'll step across one radius length at a time and for each position step up and down  by the  a radius decreased by amount of overlap with the next row [sin(30 degrees) = 0.866].
bulletDrawing a version of the board rotated by 30 degrees so that the longest diagonal becomes the longest column requires rotating each individual hex tile forward (or backward) by 30 degrees and by  interchanging the X, Y roles in the drawing algorithm from the previous so that the long  vertical column is drawn left and right  with individual tiles in each column offset vertically one tile radius and the horizontal offset becoming 0.866 of the radius.
bulletThe "puzzle hints" diagram displays solution values for one of the solutions in the 6 outside corners of the board.  The locations were found during the board creation by temporarily assigning consecutive letters as the tiles were created.  The letters in the 6 corners were then replaced solution values as the board was displayed and the other letters replaced with blanks.     
bulletWhen adding numbers or other text characters within tiles, font size can be adjusted for the tile size by setting the Canvas.Font.Height property of the TImage.  I set it to -Radius in this program making the character height 1/2 of the tile diameter.  Characters can be centered by subtracting half of the unused height (and width) from the tile center coordinates.   Canvas.TextWidth and Canvas,TextHeight methods can return those values.   

Running/Exploring the Program 

bulletDownload executable
bullet Download source   

Suggestions for Further Explorations

User play could be implemented  probably by creating an internal square board representation with cell outside of the board areas containing negative numbers, empty valid board tiles by 0 characters and filled tiles with the 18 valid tile values. 

Created: December 30, 2015

Modified: May 15, 2018

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