Problem of the Month: October and November
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Quadrilaterisation and Hexagon movesConvince 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 |