A Systematic Approach to Filling m-by-n Numerical Arrays

G. Stolyarov II
January 14, 2009
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A sample image This is an original mathematical work which has occupied much of Mr. Stolyarov’s attention in 2008. With the assistance of Dr. David Murphy, Mr. Stolyarov arrived at some new insights regarding the filling of a special kind of numerical puzzle – the m-by-n array. In the process, Mr. Stolyarov has developed a new kind of notation that makes the expression of sums of terms and compositions of summations much more concise.
The entirety of Mr. Stolyarov’s paper can be read and downloaded here.
Mr. Stolyarov also created an interactive presentation to accompany this paper. The presentation contains a variety of diagrams, examples, and illustrations of Mr. Stolyarov’s work on filling numerical arrays. If you would like to use Mr. Stolyarov’s presentation for your own lectures or simply your studies and enjoyment, you may download it here.
This paper develops a systematic way to fill any m-by-n numerical array where the row and column constraints are specified and the sum of the row constraints is equal to the sum of the column constraints. This problem has both relevance, due to the growing interest in numerical puzzles, and applications, especially to the field of algebraic geometry. In this paper, it is shown that any m-by-n numerical array with the given specifications can be filled, and a directional row-by-row filling algorithm is developed for doing so. However, many of even the simplest numerical arrays have multiple possible fillings, and we seek to arrive at a way of finding how many fillings any given array has. We develop a formula for the number of fillings for m-by-d arrays, where d is the sum of the row constraints and the sum of the column constraints, and each column constraint is equal to 1. Then we proceed to find formulas for the number of fillings for 2-by-2, 2-by-3, 2-by-4, 2-by-n, 3-by-3, and 3-by-n arrays. In the process, we develop the necessary techniques, insights, and notation to enable us to develop a formula for the number of fillings for a general m-by-n array.

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