Famous numbers on a Chessboard

Abstract

In this article it is shown how famous numbers like Pascal's triangle, the Fibonacci numbers, Catalan's triangle, Delannoy's square array, the Pell numbers and Schröder's triangle can be constructed on a chessboard with a rook, knight, bishop, king or queen. Furthermore, several new triangle sums, which are all named after chess pieces that are leapers and add up numbers according to the way they leap, are introduced. Finally a new theory of how Hipparchus, who lived around 150 BC, might have calculated his two famous numbers with the aid of a 'chessboard' is presented.

In this article it is shown how famous numbers like Pascal's triangle, the Fibonacci numbers, Catalan's triangle, Delannoy's square array, the Pell numbers and Schröder's triangle can be constructed on a chessboard with a rook, knight, bishop, king or queen. Furthermore, several new triangle sums, which are all named after chess pieces that are leapers and add up numbers according to the way they leap, are introduced. Finally a new theory of how Hipparchus, who lived around 150 BC, might have calculated his two famous numbers with the aid of a 'chessboard' is presented.