An insight into Carrom: My view
Author: Ashok G. Heda
Corresponding address:
Ashok G. Heda
Sahyog Medicals,
Kedia plots, Convent Roads,
Akola – 444 005 (M.S.)
INDIA
Phone: +91 724 2453081
Email: aaheda@gmail.com
Mathematics has never been shy of correlations. New formulae, equations and puzzles appear frequently; promising simpler, quicker, more consistent and satisfying results. Many of us invested directly, indirectly, partially, heavily or repeatedly in mathematics over the years. In reality, ‘Mathematics’ is usually an exacting, technical and rapidly developing element of our everyday practice with its emphasis on numerical correlations. I am writing this article with an intention to focus on such numerical correlations observed in Carrom, which is a very popular indoor game often enjoyed by people of all ages.
A pink queen in centre surrounded by alternate black and white draughts; the arrangement fascinated me a lot and I have started thinking on it from last couple of years. When reviewed speculatively, I found that some correlations can be drawn between areas of circles (draughts) and the space between them and can also be mathematically defined. Although not a mathematician, I have tried humorously to present my observations here. If accepted by experts, I would be highly encouraged for my small contribution to mathematics.
The routine arrangement of draughts on a Carrom board is shown below. Let queen be the central circle. Let black & white draughts arranged hexagonally surrounding it form first (n1) and second (n2) series of circles.
Statement -
“When same sized circles are arranged hexagonally around a central circle, the ratio of sum of areas of hexagonally arranged circles excluding central circle to the area of remaining space between them is 3:1 and remains constant up to nth series of hexagonally arranged circles”.
Proof -
Let radius of each small circle (draughts) is 2 units. Radius of large circle (indicated in red colour) is equal to 10 units.
Therefore,
Area of central circle (C) = Area of each small circle = 4π
Sum of areas of hexagonally arranged circles (H) = 18 (4π) = 72π (i)
Area of large circle (L) = 100π
If S is the total area of remaining space, then
L – C = P + S
S = (100π - 4π) – (72π) = 24π (ii)
From (i) & (ii), it’s demonstrated that the sum of areas of hexagonally arranged circles is proportionate to the area of remaining space between them and their ratio is 3:1. This ratio remains constant up to nth series of hexagonally arranged circles, as tabulated below.
n
|
Sum of areas of hexagonally
arranged circles
|
Area of remaining space
|
Ratio
|
1
|
24π
|
8π
|
3:1
|
2
|
72π
|
24π
|
3:1
|
3
|
144π
|
48π
|
3:1
|
4
|
240π
|
80π
|
3:1
|
5
|
360π
|
120π
|
3:1
|
While studying the above relationship, I have found another interesting thing. If the number of series of circles hexagonally arranged around a central circle is known, the total number of circles (TC) including central circle can easily be determined using a new formula as given below.
Where, x is the number of series of circles around a central circle.
e.g. if there are 4 series of circles around a central circle, as shown below
using above formula, total number of circles can easily be calculated as
