Percolation

 REMARKABLE PROPERTIES OF PAWNS ON A HEXBOARD

 

We know that on a diamond shaped hexboard, completely covered with white pawns and black pawns whatever their number, there will always exist a chain of pawns of the same color white or black, linking either the 2 extreme columns, (Diag. N°1), or the 2 extreme rows, (Diag. N°2).

Percolation - D1

It is not known however that their ALWAYS exists not one chain of pawns only, but TWO chains of pawns:

–  of different colours linking the 2 extreme columns, (Diag. N°3), or

–  of different colours linking the 2 extreme rows, (Diag. N°4), or

–  of the same colour, one of them linking the 2 extreme columns, and the other linking the 2 extreme rows, (Diag. N°5).

Percolation - D2

These more extensive properties have often been noticed and considered as singularities resulting from a particular distribution of the pawns; showing the permanence and proving it constitues a venture which had never been carried through to a succesful conclusion.

We give a direct and very simple demonstration of these new properties and in annex a demonstration by recurrence.We could also use Brouwer’s fix-point theorem, as David Gale did in 1970 to demonstrate that a chain of pawns linking 2 opposite extreme sides always existed, but we prefer to use the following reasonning by recurrece which enables us to understand the specific causes of these properties.

These new properties are equivalent to the following detailled property: there always exists either a chain of white pawns linking the 2 extreme rows, or a chain of black pawns linking the 2 extreme columns, or to the following detailled property: there always exists either a chain of black pawns linking the 2 extreme rows, or a chain of white pawns linking the 2 extreme columns. We demonstrate these last 2 properties by using Pierre de Fermat’s method of infinite descent.

We next define the notion of opposite zones on the edge of a hexboard, which enables us to extend these properties to other hexboards in the shape of a parallelogram, trapesium, rectangle or any other shape.

A chapter is devoted to hexboards in the shape of a triangle and possessing the well-known following property : there always exists a pawn linked to the 3 sides of the triangle due to 3 chains of pawns of the same colour as this pawn.

In the chapter « Other example » we reassemble the previous properties for a hexboard of any shape.

Finally, we knew that it was practically impossible to calculate the probability of obtening a chain of pawns linking the two extreme columns or the 2 extreme rows of a hexboard in the shape of, for exemple, a rectangle of 7×11 cells. Thanks to the properties we have defined in the chapter 4, this calculation becomes possible. We give some results obtained by handwritten calculations for hexboards with a small number of cells.  Although these calculations can be done by hand for larger hexboard, in this case it will be preferable to use a programme whose algorithm reproduces the exposed method of calculation.  

See the study: Propriétés remarquables