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. |
