Lattice Path Counting and Optimal Utilization of Multi-Channel Multi-Server Resources: Decorating the Math Magic Flower

Abstract

Inspired by a recent study of a Math Magic Circle [MMC], we undertake a similar study of a mathematical construct -- called Math Magic Flower [MMF]-- involving the first 81 natural numbers {1, 2, ..., 81} - each one being used exactly once. Of these numbers, only one viz., the number 41 acts as an isolate and it occupies a special position in the MMF. The geometry exhibits the number 41 holding a central position, better identified as one acting as the centre of six concentric circles. Along the circumferences of these circles are placed the natural numbers - each circle holding 10 of the numbers. As a matter of fact, the third and the fourth circles accommodate 20 numbers each. Thus we have the count 81 (= 1 + 10 + 10 + 20 + 20 + 10 + 10). The placement of the numbers is very special. There are intersecting straight lines with each line supporting exactly 3 of the 80 numbers and more over, the sum of the three numbers is the same (=123) for all lines. Our application of the magic flower deals with a navy pier operating from the central position towards outwardly navigation of ships. Or else, it could as well be in the opposite direction - from the farthest position on the sixth circle towards the centre touching each of the circles in between. There is a sense of Lattice path counting towards optimal utilization of Multi-Channel Multi-Server Resources.