Problem of the Month: October and November

Quadrilaterisation and Hexagon moves

Consider a convex 2n-gon in the plane. Divide it into quadrilaterals by drawing some diagonals which do not intersect.
Convince yourself that there are necessarily n-2 such diagonals. Call this a quadrilateralisation of the polygon.

Question How does one produce new quadrilateralisations from old ?

For instance: take any two quadrilaterals that have a common edge. The rest of their edges then form a hexagon of which this common edge is one of the three principal diagonals. In this hexagon, remove this diagonal and insert another principal diagonal. This gives a new quadrilateralisation. See figure below.

In the above figure we have two quadrilateralisations of an octagon ABCDEFGH related by a hexagon move on the hexagon ABCDEF by interchanging the principal diagonal BE by another one FC.

Problem of the month : Prove that any two quadrilateralisations of a convex 2n-gon in the plane are related by a sequence of such `hexagon moves'. ( Rotations are not allowed.)


Please submit your solutions to courses@isibang.ac.in

Previous Problem of the month contests can be found at the links below:
March and April
February
August

[Stat-Math unit] [Indian Statistical Institute]