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'Chess' Polygons (Posted on 2009-04-24) Difficulty: 4 of 5

a b c d e f g h i j
 1 
C E B F E A E B H A
A E H D F G E G D C
F E B A H H B B C F
D A D G E D H D B C
C E D E G C G C D F
C C B F A A C D G C
F B C B C C B H E C
A C E B E B A D D G
D E G B C H D E B C
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 


Draw the following set of convex polygons with vertices in ascending and consecutive alphabetical order:
triangle, square, pentagon, hexagon, heptagon and octagon.

Each vertex sequence begins with an A with subsequent (and all) vertices being 3 squares apart as defined by a chess knight's "L" shaped move or the queen's diagonal or orthogonal move.

No polygon is to have a common vertex with another. Should this occur then one or the other is not part of the solution.

  Submitted by brianjn    
No Rating
Solution: (Hide)

This was to have been my official solution (but see below):
The problem was created in Excel and a series of polygons were created using the 'Drawing' toolbar. These were moved across and down the grid, rotated as appropriate and similarly moved to test for other vertex sets.

Three "red herrings" were deliberately installed since it would be a mere exercise to begin with all of the A's in turn and test for the next vertex.
Polygons embedded:
Triangle: d3, g3,g6
Square:   f1, i4, f7, c4
Pentagon: b4, c6, e7, h4, e1
Hexagon:  g8, d8, a5, d2, g2, j5
Heptagon: e6, g7, j4, i2, g1, e2, d4
Octagon:  f6, d7, b6, a4, b2, d1, f2, g4

Known "red herrings":
          e6, c6, e8
          a8, d8, a5
          g8, i9, j7, h6
Graphically:
  a b c d e f g h i j
 1 
C E B F E A E B H A
A E H D F G E G D C
F E B A H H B B C F
D A D G E D H D B C
C E D E G C G C D F
C C B F A A C D G C
F B C B C C B H E C
A C E B E B A D D G
D E G B C H D E B C
  2  
 3 
 4 
 5 
 6 
 7 
 8 
 9 

Other than one vertex I have noted in comments that Jyqm's solution and mine were the same.

Since I had not considered all possible arrangements of polygons which could be constructed according to my rules, Charlie has revealed a range of other solutions.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re(3): computer solutions -- fixedbrianjn2009-04-25 04:32:29
Solutionre(2): computer solutions -- fixedCharlie2009-04-25 00:57:55
re: computer solutionsCharlie2009-04-25 00:43:11
Solutioncomputer solutionsCharlie2009-04-25 00:29:01
SolutionsolutionJyqm2009-04-24 17:05:39
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