# A cute bare basic proof of factorization of a^{3}+b^{3}+c^{3}-3abc by Gopal/Sravan Pithwa

This proof is not from Titu Andreescu. But original attempt by student Gopal/Sravan Pithwa. It had never occured to me a brute force proof would be so easy in this case :_-) Surely we learn a lot from children/students !!!

Prove that $a^{3}+b^{3}+c^{3}-3abic = (a+b+c)(a^{2}+b^{2}+c^{2}-ab-bc-ca)$

Proof:

LHS = $a^{3}+b^{3}+c^{3}-3abc = (a^{3}+b^{3})+ c^{3}-3abc$ $= (a+b)(a^{2}+b^{2}-ab) + c^{3}-3ab$ $= (a+b)(a^{2}+b^{2}-ab+3ab-3ab) + c^{3}-3abc$ $= (a+b)(a^{2}+b^{2}+2ab -3ab)+c^{3}-3abc$ $=(a+b)((a+b)^{2}-3ab)+c^{3}-3abc$ $= (a+b)^{3}-3ab(a+b)+c^{3}-3abc$ $= (a+b)^{3}+c^{3}-3ab(a+b)-3abc$ $=(a+b+c)((a+b)^{2}+c^{2}-c(a+b))-3ab(a+b+c)$ $=(a+b+c)(a^{2} + b^{2}+c^{2}+2ab-ca-bc) - 3ab(a+b+c)$ $= (a+b+c)(a^{2}+b^{2}+c^{2}-ab-bc-ca)$.

QED. Hence, $a^{3}+b^{3}+c^{3}-3abc = (a+b+c)(a^{2}+b^{2}+c^{2}-ab-bc-ca)$

# Fibonacci Numbers: Recreational math

Reference: ICEEM Year 8 Math text book, Australian Mathematical Society.

A Fibonacci sequence is a sequence $F_{1}, F_{2}, F_{3}, \ldots$ of numbers in which each term from the third one onwards is the sum of the two terms that immediately precedes it. You have to have two numbers to start with, $F_{1}$ and $F_{2}$. These are called seeds. Then, $F_{3}=F_{1}+F_{2}$ $F_{4}=F_{2}+F_{3}$

and so on. The classic Fibonacci sequence has 1 and 1 as its seeds. It first ten terms are:

1,1,2,3,5,8, 13, 21, 34 and 55.

Use a calculator when appropriate in the following:

Activity 1

Write out the classic Fibonacci sequence as far as its 25th term, $F_{25}$. Before you calculate $F_{11}$, make a rough guess of what $F_{25}$ will be. See how good your guess turns out to be.

Activity 2

Pick any two numbers as seeds and work out the first 20 terms for that Fibonacci sequence. Pick entirely different seed numbers from the person besides you (example, your friend, or teacher, or any one else imaginary :-)), and keep your list reasonably neat, as we will be coming back to it in a little while.

Activity 3

Swap your two seed numbers from Activity 2 around and figure out the first twenty terms in the new Fibonacci sequence. (If, your sequence in Activity 2 started 6, 11, 17, 28, 45, …) your new sequence will start 11, 6, 17, 23, 40, ….). Yes, you do get quite different numbers from Activity 2.

Activity 4:

It is now time to make a few observations about your Fibonacci sequences.

• The classic sequence in Activity 1 has two odd seeds. This gives a certain pattern of odd and even terms through out the sequence. What happens if you start with two even seeds, or an odd and even seed? Explain.
• Compare the 10th terms you generated in each of the sequences in Activity 2 and Activity 3. Which one is larger? Compare the 20th term as well. Can you explain what is happening?
• Use the calculator to divide the term $F_{10}$ by the term $F_{9}$ immediately before it. Write your answer down. Then, do the same with the second sequence. Now repeat the calculation with terms $F_{20}$ and $F_{19}$. Do you notice anything interesting? Did any of the “other people” who are doing this activity get the same ratio? They should all have got the same ratio. They should all get the same number, even though there may be v small diferences in the sixth decimal places.
• For the classic Fibonacci sequence, the first two terms larger than 1000000 are $F_{32}=1346269$ and $F_{33}=2178309$. Use these two values to see if what you noticed in the previous ratio calculations still holds for higher order terms in the classic Fibonacci sequence.

The number you obtained (to a good approximation) in the ratio calculations is famous and interesting enough to deserve its own Greek letter. It is called $\phi$ (pronounced to rhyme with spy). It is called the golden ratio or golden mean. It is a very interesting number with long history.

Search Google and you will discover some amazing facts about $\phi$. It appears in many different ways in geometry and architecture.

Now try calculating these values and see what you notice about them:

a. $\phi^{2}$

b. $\frac{1}{\phi}$

c. $(2\phi -1)^{2}$

Regards,

Nalin Pithwa.

# A little big announcement from National Museum of Math, NYC

Dear MoMath families,

There’s always more to learn at MoMath!  The Museum’s pre-K through 12 programming reveals a world of engaging mathematics and offers fun, hands-on activities your child can do right at home, led by classroom-seasoned, top-notch educators.

MathPlay, MoMaths program for preschoolers
Whether your preschooler is just learning to count or is gearing up for kindergarten, MathPlay will instill a love of mathematics in each child through educational games, catchy songs, and intriguing problem-solving challenges.  mathplay.momath.org

“It was the first time my daughter saw the use of a number line for addition and she really enjoyed the class.  She was very focused for the full 30 minutes thanks to our engaging instructor who walked her through step by step.” — a parent writing about MathPlay

Student Sessions for grades pre-K through 12
Join young math fans from around the world for one of MoMath’s 45-minute, interactive Student Sessions, such as the ever-popular “Shape Shifters” or “Breaking Codes,” or the newest session, “Color Grids.”  In “Color Grids,” students generate their own infinite patterns while exploring simple cellular automata.  studentsessions.momath.org

“My son spent the weekend continuing to play with his shapes and his interest in the content was elevated due to this session.  Zoom was an excellent platform and the educator was very engaging.  Excellent session!” — a parent writing about MoMath Online: Student Sessions

Now you can share the gift of math with friends!  Gift registrations for Student Sessions are available at mathgift.momath.org.

For more mathematical fun, be sure to check out MoMath’s complete lineup of upcoming adult events and children’s programs at events.momath.org.Regards,
National Museum of Mathematics.

PS: I am sharing the email content I received for the benefit of Indian students….Hope the awareness spreads and Indian kids enrich their love and knowledge of math 🙂

# Bill Casselman’s Euclid: thanks to ClayMath

The purpose is only to share and spread the awareness of availability of this second master piece on Euclid. Thanks to Clay Math Organization for serving students world wide, and thanks to the generous Mr and Mrs Clayton. I hope my math olympiad students will enjoy this and enrich themselves mathematically.

http://www.math.ubc.ca/~cass/euclid/

# Wisdom of V. I. Arnold, immortal Russian mathematician

Development of mathematics resembles a fast revolution of a wheel: sprinkles of water are flying in all directions. Fashion — it is the stream that leaves the main trajectory in the tangential direction. These streams of epigone works attract most attention, and they constitute the main mass, but they inevitably disappear after a while because they parted with the wheel. To remain on the wheel, one must apply the effort in the direction perpendicular to the main stream.

V I Arnold, translated from “Arnold in His Own words,” interview with the mathematician originally published in Kvant Magazine, 1990, and republished in the Notices of the American Mathematical Society, 2012.

# Two semantic paradoxes

1. I am lying
2. The only statement on this whiteboard is false.

Whereas Russell s paradox in set theory is mathematical.

Regards

Nalin Pithwa