A nonlinear rise in Team India's fortunes is inevitable #Ganita

 Three events reported in a day:

https://twitter.com/niravstoons/status/1759205630274089143 

BAI Media (@BAI_Media) / X (twitter.com)

https://twitter.com/AdvAshutoshBJP/status/1759215880679141779  

A 9-year old boy chess prodigy, a 17-year old girl's nerveless display to clinch Badminton gold for India to win the Asian team championship, a 22-year old cricketer's effortless double century to swing a cricket test and series in India's favor. All in a day's news. Such stories were also celebrated in the past but infrequently and were treated as big surprises, but three stories reported in a day is no longer unexpected. The potential size of India's sports base is such that successful outcomes become inevitable if the number of opportunities to convert to a podium finish is increased even by a small percentage.  Even though a specific outcome depends on individual karma and their sadhana and is not deterministic, the simple Ganita of outcomes as a group is that:

Given 'n' opportunities and a probability 'p' of success per opportunity, the expected number of successful outcomes = np. 

For a nation, n represents the size of its sporting base: the number of individuals with a liking for the outdoors, are athletically gifted AND their family is willing to consider a brief or long career in or through sport.  On the other hand, p captures the complex impact of identifying such talent and the dharma of their selection criteria, diet, training, exposure, and ultimately, their conscious mental ability in the heat of battle.

Until a decade ago, both opportunities (n) and the chance of success (p) were low. The current Indian state is that both n and p are low in comparison to natural sporting nations like Australia. Every small improvement in India's health and fitness, sports and education to reward, encourage, and nurture sporting talent, will simultaneously raise n and p and yield a nonlinear and sustainable increase in the number of such success stories in Indian sport. 

For example, a modest 10% improvement in n and p, improves the expected number of successes to (1.1n * 1.1p) = 1.21 (np), i.e., a 20% increase in the expected number of successes.

The #Ganita of Sandwich Slicing: Are Triangles better than Squares?

Question- How do you like your sandwiches sliced: Triangular or Square?

Triangle

Picture credit: (VIDEO) Tricolour Triangle Tea Sandwiches (theinspirationalnook.com)

Square

(Picture credit: Ribbon Sandwiches (Rainbow Sandwiches) - The Flavor Bender)

To answer this from a Ganita perspective, let's apply some high-school Kshetraganita (the original geometry):

Consider a square bread slice of side length '2a' (area 4a²). Slice it into 4 pieces of equal area a² each in two different ways:

1. 4 squares of side length 'a':

area of each square = a², perimeter of each square = 4a = 2 × 2a, i.e, twice the length of the original side.

2. 4 right-angled triangles:

Confirm the area of each triangle (crust-edge as base) = ¹⁄₂ base × height = ¹⁄₂ × 2a × a =  a²

Next, apply the 'square on the diagonal' result from the Sulbasutras to calculate the length of the diagonal on the original square:

= √(4a^2 + 4a^2) = 2√2 a.
The sides of each triangle are half as long as this diagonal.

Alternatively, using the 'circlometry' of Aryabhata (the original trigonometry), we let φ = the corner angles of the 4 right angled-triangles = 45°.

The original jya of Aryabhata calculates length and not a ratio like sine (it is of the form "R sine φ", where R = radius of a circle), we can obtain the length of the side = 2a×sin(φ) = √2a 

 From this, we get the triangle's perimeter

= base + 2 sides 

= 2a (1 + √2) ~ 2.414 times the length of the original side

The perimeter of the triangular slice is about 21% longer than that of the square slice of the same area. What sorcery is this :)

Since the area of the sandwich filling that is visible to the consumer = perimeter × thickness of the filling,  triangular slices may have better curb appeal as they show off the filling 21% better than square slicing, which is itself twice as good as not slicing at all. 

Each square slice has one full crusty corner, and a crust length of 2a. The triangle is tied on crust length but has two 'half-corners'. So if your kid is fussy about corners, triangles may prevent left-overs in the lunch box. If they don't like crust-edges and the fillings are not exciting, then you may need to go back to the square one.