Another formula and other stuff

The latest formula discovered by our friend is

N^k - (N-1)^k = Sum_r=0^k-1 N^r (N-1)^{k-1-r}

I asked him how he found it, he said that he had observed this for small k when he was young (well he is 8 now!), and generalized it to all k recently.

He has also started learning about logarithms. I asked him for the value of log 2.5. His first question was -- to what base? I said 10. After a few seconds he said 0.4. I assumed that he made a guess, he probably knew that log 2 = 0.3, and log 3 is close to 0.5. After a few minutes I asked him how he got it --- he said that he knew that 4^5 = 1024, so 5 log 4 = 3, so log 4 = 0.6. Now log 2.5 = log 10 - log 4 = 0.4. Pretty cool! I asked him for logs of some more numbers, each time he came up with a different way to evaluate it, so he not only knows what a log is, but understands it at a deep conceptual level.

A cool discovery

Our young friend, all of 6 years old now, made a cool conjecture: for all k, for all n >= k, the last k digits of 2^n are the same as those of 2^(n + 4 * 5^(k-1)), and that this is not the case for any power of 2 between these two. He verified this for k = 1 and 2, and then guessed and verified it for k=3, which is truly amazing, as his conjecture would then be that the last three digits of 2^103 are 008. This is not particularly easy to verify! (Note that he does not know enough algebraic notation to write it out this way, but he came up with the expression all by himself.)

The conjecture can be proved using some simple number theory, but for a 6 year old to discover this ...

Other interesting problems he solved before: Find the number of integers between 0 and 1000 that can be written as the difference of two squares? Ans: 751.
What numbers with all digits repeated are primes? He figured out that the only non-trivial such primes must have the digit 1 repeated a prime number of times.
Of course these problems seem trivial in the light of the above conjecture, but I was very impressed at the time. He has also started reading Recreations in the Theory of Numbers by Albert Beiler, and likes Raymond Smullyan's logic puzzle books.

Recommendations for Math Prodigies

This blog originated from a visit to a friend's place. His son, who is not yet 6, is very interested in math (he was writing out the Fibonacci series on a sheet of paper when I saw him). I asked him to sum up the numbers from 1 to 100 (like Gauss), and to my surprise, he gave an answer in a couple of minutes --- he had inferred the right idea on summing an arithmetic progression, though he made a slight error in the calculation, but it was still incredibly impressive. We then asked him to name two primes that sum to 19, he immediately replied 2 and 17. Then we asked him for another pair, he asked me if I was sure there was another, saying that there were no others as 2 is the only even prime.

So it was clear that this kid is unusually gifted in mathematics. I asked my colleagues how we should nurture his interest further, without pressuring him in anyway, but just pique his curiosity about numbers further (and he is very curious indeed!). This is a summary of their responses. I will keep adding more ideas, please send them to vineetgupta AT gmail DOT com.

Interesting books to read: In many cases these books are too difficult to read for a 6 year old, so the parents would end up reading the books and then talking to the child.

Recreations in Number Theory - Beiler
The Lore of Large Numbers - Davis
Martin Gardner's Mathematical puzzle books. Aha! Insight.
Adventures in Mathematics
Mathematician's Delight - W W Sawyer
Sideways Arithmetic from Wayside School - Sacher
What is Mathematics? - Courant and Robbins. For older kids.
Smullyan's logic puzzle books : What is the name of this book?, Alice in Puzzleland, The Lady or the Tiger (inspired by Frank Stockton's beautiful story), The Riddle of Scheherazade, Forever Undecided.
Asimov's Realm of Algebra
Weeks' Shape of Space
The Number Devil
How to solve it - Polya
Flatland - Abbott
John Allen Paulos' books.
Proofs from the Book
The man who counted.
The Penguin Dictionary of Curious and Interesting Geometry

Software, tools, websites:
POV-ray an open source ray tracing engine for geometric intuition, coding etc.
squeakland.org (programming).
Happy Cube
Soma cube(you can make soma cubes at home)
Wire puzzles


Math Programs:
Math summer camp Ross at Ohio state
PROMYS at Boston U after a few years.
EPGY Stanford.
Find a local Math Circle (Ask Tom Davis www.geometer.org ).
The Gelfand Program for talented and inquisitive students.
Arrange a visit MSRI Berkeley and chat with mathematicians.
Math circles at Stanford and Berkeley

Math areas to focus: (These are not usually taught in schools).
Number Theory
Graph theory
Combinatorics
Set Theory and set theoretic definitions of numbers, arithmeic etc.
Abstract Algebra
Mathematical puzzles like Sudoku (solve and later build).
CS ideas like binary search, sorting algos, dfs and bfs, probability computations, optimal algorithms for games like Mastermind)

Interesting questions to ask and encourage discovery:
Find Euler's tour in a graph
Ratio of circumference to diameter
Find rules for divisibility in different bases ( e.g. for 3 and 5 in base 16)
Limits and infinite series 1 + 1/2 + 1/4 + ....
Convergence of sequences
Show countability of rationals and uncountability of reals to demonstrate different kinds of infinity
Show that there are infinitely many primes.
Show that sqrt(2) is irrational.
Given the numbers 1 through n, choose n/2 + 1 numbers. Show that this set must contain two relatively-prime numbers.
Show that the sum of odd numbers from 1 to 2n+1 equals (n+1)^2.

Other ideas:
Don't focus only on math - let him be a kid - play soccer, build sandcastles etc.
Focus on things not taught in school.
Do not hold him back!
Talk to teachers and find out about techniques used for kids so far above norm. Do not force him to sit through standard math classes as he already knows this stuff.
Find peers who share his affinity to numbers for interaction.
Chess clubs for peers.
Leave lots of fun books lying around, especially a math and science encyclopedia.
Do other fun activities that apply math: build a robot, perform experiments with chemicals and electricity, build a radio, program a computer game, build a tree house etc.
Let him invest for his education.
Find a smart caring adult to talk to him and generally goof off about math.