### 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.

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.

## 2 Comments:

Excellent compiled list.I don't know if my kids will go through these books or concepts but I am going to go through them myself. It would be interesting to know in each of the cases what made the kid a prodigy - is it genetics or environment or something else.

Vineet, tanks for a very interesting blog, and links to good math books.

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