My Art

The Book Empire Strikes Back

This is my updated book list Starting from June 2018
1.   7^th Heaven (James Patterson)

2.   The Book Jumper (Methchild Glaser)

3.   Listen to your heart (Kasie West)

4.   Nice and Mean (Jessica Leader)

5.   Save the Date (Morgan Matson)

6.   Warriors: SkyClan’s Destiny (Erin Hunter)

7.   Pivot Point (Kasie West)

8.   Warriors: A Vision of Shadows- Shattered Sky (Erin Hunter)

9.   Pretty Little Liars: Flawless (Sara Shepard)

10.      Warriors: Crookedstar’s Promise (Erin Hunter)

11.      Little Monsters (Kara Thomas)

12.      Postcards from Venice (Dee Romito)

13.      The Last to Let Go (Amber Smith)

14.      Warriors: A Vision of Shadows- Darkest Night (Erin Hunter)

15.      To All the Boys I’ve Loved Before (Jenny Han)

16.      Pretty Little Liars: Perfect (Sara Shepard)

17.      The List (Siobhan Vivian)

18.      Three Dark Crowns (Kendare Blake)

19.      Royals (Rachel Hawkins)

20.      Catwoman: Soulstealer (Sarah J. Maas)

21.      Losing Brave (Bailee Madison & Stefne Miller)

22.      Love, Life, and the List (Kasie West)

23.      The Clique (Lisi Harrison)

24.       Pretty Little Liars: Unbelievable (Sara Shepard)

25.      Warriors: A Vision of Shadows- River of Fire (Erin Hunter)

26.      That's Not What Happened (Kody Keplinger)

27.      Warriors: Tigerheart's Shadow (Erin Hunter)

28.      This Savage Song (Victoria Schwab)

29.      Legendary (Stephanie Garber)

30.      Warriors: Crowfeather's Trial (Erin Hunter)

31.      The Winner's Curse (Marie Rutoski)

32.      One of Us is Lying (Karen M. McManus)

33.      Lucky in Love (Kasie West)

34.      The Cheerleaders (Kara Thomas)

35.      Once and for all (Sarah Dessen)

36.      Lady Midnight (Cassandra Clare)

37.      An Ember in the Ashes (Sabaa Tahir)

38.      Red Queen (Victoria Aveyard)

39.      Reboot (Amy Tintera)

40.      Warriors: Tallstar's Revenge (Erin Hunter)

41.      Renegades (Marissa Meyer)

42.      Throne of Glass (Sarah J. Maas)

43.      Pretty Little Liars: Wicked (Sara Shepard)

44.      A Study in Charlotte (Brittany Cavallaro)

45.       

Pigeon-Hole Principle

What is a Pigeon-Hole Principle?

If I ask you the following questions "Are there two people in NYC who have the same number of hairs on their head?" or  "Are there two homeschoolers in Long Island who have the same birthday?"
What will you answer?  The answer to both questions is "YES".
How?   The proof uses the Pigeon-Hole Principle a simple concept yet a powerful tool.

Lets say that there are 6 pigeons in a cold winter day and they have to shelter in 5 holes. If all pigeons should have shelter, then at least two pigeons have to fit in a hole. This is generalized as the pigeon-hole principle.If there are x pigeons and y holes, and x>y and each hole must be occupied, at least two pigeons will have to fit in a hole.

This looks like common knowledge.  Why is it important in mathematics to be called a  "Principle"?  

Let us look at how we will answer the questions asked above?  A human on an average has 150,000 hairs.  There are around 2 million people in NYC.  Think of the number of hairs as holes and people as pigeons.  So by Pigeon Hole Principle there are at least two people who have the same number of hairs.

I read the following proof recently in the book  A Walk Through Combinatorics and it really fascinated me. 
Look at the sequence: 7, 77, 777, 7777,...
Claim: There is an element of the sequence which is divisible by 2003.  

How to prove?  By Contradiction.

On the contrary that none of the first 2003 terms are divisible by 2003.  The possible remainders are 1,2,3,...2002.

There are 2003 members of the sequence.  They are like pigeons.
There are 2002 remainders.  They are like holes.

By Pigeon-Hole principle, there should be at least two pigeons (two members of the sequence) in a home (remainder).
Thus at least two members share the same remainder.

Name the two numbers of the sequence  as x and y.

x = d(2003) + r
y = f(2003)  + r.

Since x and y are in the sequence, we can assume that x > y.

x-y = (d-f)2003 

This implies that x-y is divisible by 2003.

If we show that x-y is a member of the sequence 7, 77, 777,... then we have showed that our assumption is false.  

But why should x-y be of the form 77...7

We know that x = 7777...777
                        y =       7...777
Hence x-y = 777 x 

Since 2003  divides x-y, and 2003 does not divide 
2003 must divide 77...777 < x.

Hence we have found a number in the sequence that is divisible by 2003 which renders our assumption false.  Hence 2003 divides at least one of the first 2003 terms.

Is 1729 really a boring number?

G.H Hardy,  a famous  number theorist, also mentored the Indian  mathematician Srinivasa Ramanujan.

Once, when Ramanujan was at a hospital in Putney, Hardy visited him. Hardy rode a taxicab numbered 1729 (came to be called as Hardy-Ramanujan number) on the way there, and he remarked that 1729 was a dull number. 1729 was really not boring at all!  Ramanujan instantly replied to Hardy that it was the smallest number to be written as a sum of two positive cubes in two different ways.
1729 = 1³ + 12³ = 10³+11³

But it also has a few other interesting properties!

• 1729=7*13*19, a product of three primes. Any number that is a product of three primes is called a Sphenic number.

• The three primes 7, 13, 19 satisfy 13=1*7+6  and  19=1*13+6. Such a number is a Zeisel number.

• 1+7+2+9=19, 19*91=1729.  This property was discovered by Masahiko Fujiwara.

Why is square root of 2 irrational?

In this and the next few blogs I will write proofs of irrationality that I enjoyed learning recently.  I will start with the simplest one on the proof of irrationality of square root of 2.

Have you ever wondered how many primes are there.

TA for Calculus BC class

I enjoy being a TA for the Calculus BC and AB class this year.  I never imagined that I will get to be a TA at 13.  I did the course for my AP Calc BC last year.

More Logic Puzzle Records

I had made a few more records.  I finally found time to put it in my blog

The Cat(Sonnet 156)

The cat has come outside the door;
her pelt glows dark into the night.
While thinking about having more,
the stars will shine so bright.
She is prowling through the fog,
Stalking without a trace through the cold mist.
She is being chased by a dog,
but oh, the dog has missed.
She is panting in a cave with relief;
a cave where all sounds echo.
Her paws spring upon a leaf;
she will never let go.
At the moment the sun will rise,
she does not want a surprise.

Sonnet 155 - My First Sonnet

Our love once grew strong,

but now it is starting to fall.

We thought it would last long,

but it did not stay at all.

I thought that looking in your eyes

 tasted like sweet food,

but then I heard all the lies

and knew that you were no good!

I thought that we were forever

 when you  first saw me,

but love now feels like never,

can't you see?

Maybe you should find a heart that is not mine,

and then all of your love could shine!

Cosmetic testing on animals-cruelty

I support a ban on testing cosmetics on animals not only because there are synthetic alternatives for testing maquillage, but also because animals cannot provide consent and human reactions to chemicals are not the same as those of animals.

What writing means to me

For me, writing has a completely new definition. I use it as a power, weaving together exotic sentences that soon join together to form legendary artwork. When I see my writing, I now see bright, beautiful words. But it was not always that way.

I used to hate writing. It was a force commanded against me. No matter how hard anyone tried to move me from this place, I stubbornly would not budge. But two years ago, I took a writing course that changed my life forever. From that moment on I could never stop writing in various formats. I noticed that I have an affinity for writing short stories and poems. More recently, I took an essay writing course for the SAT. This course will take me one step further.

When I use writing as my own force, wild words flow right out of my hand. The more power I use, the more glamourous the passage turns out to be. When I am done, all of the paragraphs appear to shine brightly into my eyes.

And even later in my life, my writing will get me into top-level colleges and may even earn me widespread fame. In conclusion, writing is a flame that I burn brightly with.