Pick’s theorem:
Consider a square lattice of unit side. A simple polygon (with non-intersecting sides) of any shape is drawn with its vertices at the lattice points. The area of the polygon can be simply obtained as 
Do you like this challenge?
Nalin Pithwa.
There are some pieces of mathematical folklore that you really should be reminded about, even though they are “well-known” — just in case. An excellent example is the Mobius band.
Augustus Mobius was a German mathematician, born 1790, died 1868. He worked in several areas of mathematics, including geometry, complex analysis and number theory. He is famous for his curious surface, the Mobius band. You can make a Mobius band by taking a strip of paper, say 2 cm wide and 20 cm long, bending it around until the ends meet, then twisting one end through 180 degrees, and finally, gluing the ends together. For comparison, make a cylinder in the same way, omitting the twist.
The Mobius band is famous for one surprising feature: it has only one side. If an ant crawls around a cylindrical band, it can cover only half the surface — one side of the band. But, if an ant crawls around on the Mobius band, it can cover the entire surface. The Mobius band has only one side.
You can check these statements by painting the band. You can paint the cylinder so that one side is red and the other is blue, and the two sides are completely distinct, even though they are separated by only the thickness of the paper. But,, if you start to paint the Mobius band red, and keep going until you run out of band to paint, the whole thing ends up red.
In retrospect, this is not such a surprise, because the 180 degrees twist connects each side of the original paper strip to the other. If you don’t twist before gluing, the two sides stay separate. But, until Mobius (and a few others) thought this one up, mathematicians didn’t appreciate that there are two distinct kinds of surface: those with two sides and those with one side only. This turned out to be important in topology. And, it showed how careful you have to be about making “obvious” assumptions.
There are lots of Mobius band recreations. Below are three of them:
The Mobius band is also known as a Mobius strip, but this can lead to misunderstandings, as in aLimerick written by a science fiction author Cyril Kornbluth:
A burleycue dancer, a pip
Named Virginia, could peel in a zip,
But she read science fiction
and died of constriction
Attempting a Mobius strip.
A more politically correct Mobius limerick, which gives away one of the answers, is:
A mathematician confided,
That a Mobius strip is one-sided,
You’ll get quite a laugh
if you cut it to half,
For it stays in one piece when divided.
Ref: Professor Stewart’s Cabinet of Mathematical Curiosities.
Note: There are lots of interesting properties of Mobius strip, which you can explore. There is a lot of recreational and pure mathematics literature on it. Kindly Google it. Perhaps, if explore well, you might discover your hidden talents for one of the richest areas of mathematics — topology. Topology is a foundation for Differential Geometry, which was used by Albert Einstein for his general theory of relativity. Of course, there are other applications too…:-)
— Nalin Pithwa.
Practice Quiz:
You are most welcome to share your answers,
Nalin Pithwa
Problem:
Of all triangles with a given perimeter, find the one with maximum area.
Solution:
Consider an arbitrary triangle with side lengths a, b, c and perimeter 
Now, the arithmetic mean geometric mean inequality gives
Therefore,
where inequality holds if and only if 
Thus, the area of any triangle with perimeter 2s does not exceed 
More later,
Nalin Pithwa
PS: Ref: Geometric Problems on Maxima and Minima by Titu Andreescu et al.
Euclid alone
Has looked on Beauty bare. Fortunate they
Who, though once only and then but far away,
Have heard her massive sandal set on stone.
— Edna St. Vincent Millay (1923)
![\sqrt[3]{s(s-a)(s-b)(s-c)} \leq \frac{(s-a)+(s-b)+(s-c)}{3}=\frac{s}{3}](https://s0.wp.com/latex.php?latex=%5Csqrt%5B3%5D%7Bs%28s-a%29%28s-b%29%28s-c%29%7D+%5Cleq+%5Cfrac%7B%28s-a%29%2B%28s-b%29%2B%28s-c%29%7D%7B3%7D%3D%5Cfrac%7Bs%7D%7B3%7D&bg=ffffff&fg=000&s=0 "\sqrt[3]{s(s-a)(s-b)(s-c)} \leq \frac{(s-a)+(s-b)+(s-c)}{3}=\frac{s}{3}")
