Friedrich-Schiller-Universität Jena
Fakultät für Mathematik und Informatik
Institut für Angewandte Mathematik

# Symmetries in No Three in Line Configurations

The single letter n always denotes the size of a given grid.

## Symmetry Notation

  ===========================================================================
Symmetry classes in no-three-in-line   or   the  Group D4 of the n x n grid
===========================================================================

1-fold     2-fold      4-fold      8-fold
---------------------------------------------

reflection in mid-pendicular:               ort1        ort2

1/k rotation about center:       iden       rot2        rot4        full

reflection in long-diagonal:                dia1        dia2



## Symmetry Conditions

Symmetry class == ort1 || ort2 || full and n odd
The mid row == symmetry row must contain 2 markers. But this forces in the corresponding columns an odd number of markers. A contradiction. Therefore in ort1 || ort2 || full there exists no solution if n is also odd.
Symmetry class == rot4 and n odd
There must be a total of 2n markers which is not dividable by 4. But each quarter of the grid which must contain (2n)/4 markers. A contradiction! Therefore if n is odd in class rot4 there are no solutions.
Symmetry class == dia2
If n is even either on both long diagonals are 2 markers, or on neither long diagonals is any marker. Empirically is found that only the first case is realized! If n is odd, then on exact one of the long diagonals are 2 markers
Symmetry class == dia1
on the reflecting long diagonal must be 0 or 2 markers.
Symmetry class != iden && != dia1 and n odd
the central point can not be marked because the total number of markers must be even.
|| denotes the logical or
&& denotes the logical and
= denotes the equality relation
!= denotes the inequality relation

## Symmetry Effects

In symmetry class dia1 or dia2 a marker on a reflecting long diagonal forbids any further marker on the diagonal orthogonal crossing his position.
In symmetry class rot4 there appear interesting number theoretic phenomena. Each marker which is set in a possible configuration forbids certain other positions of the grid because further markers together with their 3 symmetry markers block certain lines. The number variates not only with the grid size but strongly depends on the actual position of the marker. E.g. look at this 10 x 10 example:
. . . . . . + . . .
. + . . o . . . + .
. . . . . . . . . .
+ . . . . . . . . .
. . . . . . . . o .
. o . . . . . . . .
. . . . . . . . . +
. . . . . . . . . .
. + . . . o . . + .
. . . + . . . . . .

The markers set at the o positions block the + locations. In general a marker can block not more than n other positions.

## Near Symmetry Misses

The size of a miss of a configuration C in regard to a symmetry class SYM let be defined as: | { SYM(x,y) | (x,y) \in C but SYM(x,y) is \notin C } |.
Due to an idea of John Selfridge I looked for nearest missed symmetric configurations. There are some which missing reflection in a(both) mid-perpendiculare:
n=3  missing ort2:  x010212
n=7  missing ort2:  :13062415240635
n=7  missing ort2:  :13240615062435
n=9  missing ort1:  .135628044707285613
n=11 missing ort2:  :372845190A460A19562837
n=13 missing ort2:  :0C4839562A1B571B2A6739480C
n=13 missing ort1:  .16094A578B2C132C8B574A0936

These are nearly missing quarter rotation symmetry of the grid:
n=3  missing rot4:  x010212
n=9  missing rot4:  :341613680802572745