Tom Körner's "The Pleasures of Counting" ( https://www.amazon.com/Pleasures-Counting-T-W-K%C3%B6rner/dp... ) is an amusingly written yet intellectually challenging introduction to applied maths for the general audience. (His website https://www.dpmms.cam.ac.uk/~twk/ will give you an idea of the writing style.)

My son (an occasional Hacker News participant) was in a similar position a few years ago. I'll list here a variety of books that should be helpful, some of which were recommended to him by summer program instructors, and others of which were recommended to me by parents of other children with similar activity backgrounds.

For access to a lot of mathematical concepts at a reasonable reading level, not at all expensive, I recommend Concepts of Modern Mathematics by Ian Stewart.

http://www.amazon.com/Concepts-Modern-Mathematics-Ian-Stewar...

Ian Stewart is a mathematician who loves to write popular writings on mathematics, and you can hardly go wrong with anything he has written.

From Zero to Infinity: What Makes Numbers Interesting by Constance Reid

http://www.amazon.com/Zero-Infinity-Makes-Numbers-Interestin...

is very accessible and covers a number of interesting topics.

The Art of Problem Solving by Richard Rusczyk and Sandor Lehoczky

http://www.artofproblemsolving.com/Store/viewitem.php?item=p...

http://www.artofproblemsolving.com/Store/viewitem.php?item=p...

is a straight-up contest preparation book, in two volumes, that your son may find interesting. Volume 2 is for high school level contests.

For an interesting (in places laugh-out-loud funny) book about the place of mathematics in modern life and how mathematicians think about mathematics, I recommend The Pleasures of Counting by T. W. Körner.

http://www.amazon.com/Pleasures-Counting-T-246-rner/dp/05215...

This one is more challenging as to reading level and as to mathematical level than the recommendations above, but well worth having around the house.

About a decade ago, when I was studying the history of mathematics, I noticed that in 1776, the world's greatest mathematician (Leonhard Euler) was in St. Petersburg, Russia, just when many of the world's greatest political scientists were either in Philadelphia, Pennsylvania (the signers of the Declaration of Independence) or in various parts of Britain (e.g, Edmund Burke and Adam Smith). To this day, the Russian-speaking world exceeds the English-speaking world in the quality of its primary and secondary mathematics instruction, and it is perhaps no accident that the first of the Clay Millennium Prize problems

http://www.claymath.org/millennium/

was solved by a mathematician who was educated in Russia. But also to this day, the United States and Britain enjoy an astonishing degree of political and economic freedom and rule of law

http://www.freedomhouse.org/template.cfm?page=594

and gain many of their best mathematicians and mathematics educators as immigrants from non-English-speaking countries. It is too early to say whether a lot of engineering-trained persons in government is mostly a feature or mostly a bug. I wish China well in going the direction of Taiwan (another place long ruled by technocrats) in developing the rule of law and an open political system with many guarantees of personal liberty. But it is by no means an invariant characteristic of human societies that those with the best math and science minds thrive best over the long term.

P.S. You did see below the fold on the submitted article, didn't you, what the blog author thinks China can count on just from the fact of the educational background of its leaders? Not much, just from that fact.

P.P.S. to respond to first reply: It's my understanding that the government of the Federal Republic of Germany consciously DE-emphasized technical education after World War II in favor of more emphasis on humanities and social science in the primary and secondary school curriculum. I thought it would trigger a mention of Godwin's Law

http://en.wikipedia.org/wiki/Godwin%27s_law

if I brought this up at first, but I've read that many observers of prewar Germany under the Third Reich looked at the quality of the scientists there (very high indeed) and thought that Germany would be hard to beat in the war. It is well known to people who read interesting histories of World War II, such as mathematician T.W. Körner's book The Pleasures of Counting,

http://www.amazon.com/Pleasures-Counting-T-W-K%C3%B6rner/dp/...

that there was a battle of scientists versus scientists in the war to find smart methods for fighting the other side. Ultimately, despite the great advantage that German's prewar primary and secondary schools and universities and civil service system gave Germany in building up a supply of smart technocrats, the Nazis' disregard of personal liberty drove away many of Germany's best scientists (notably, many Jewish scientists) and added talent to the Allied side.

If you like stories like this, there are a number of similar stories (e.g. this one, cracking the enigma code, protecting convoys) in a book called The Pleasures of Counting: http://www.amazon.com/Pleasures-Counting-T-W-K%C3%B6rner/dp/... - it's not all world war 2, but is very interesting and does contain a lot of information on the math used during the war.

A really exciting book about this phase of the war, which will be of interest to any math-oriented hacker, is The Pleasures of Counting by T. W. Körner.

http://www.amazon.com/Pleasures-Counting-T-W-K%C3%B6rner/dp/...

This is a review of said book, which is here: http://www.amazon.com/Pleasures-Counting-T-W-Körner/dp/05215...

Some favorites about mathematics:

Concepts of Modern Mathematics by Ian Stewart

http://www.amazon.com/Concepts-Modern-Mathematics-Ian-Stewar...

Numbers and Geometry by John Stillwell.

http://www.amazon.com/Numbers-Geometry-John-Stillwell/dp/038...

The Pleasures of Counting by T. W. Körner

http://www.amazon.com/Pleasures-Counting-T-W-K%C3%B6rner/dp/...

Mathematics: A Very Short Introduction by Timothy Gowers

http://www.amazon.com/Mathematics-Short-Introduction-Timothy...