Puzzles and Fractals

Today I was asked a simple but innocent question: 'If a number plus 4 is equal to six times the sum of its digits and the number plus eighteen is equal to the number obtained by reversing its digits then find the number.'

Evidently, this is neither simple nor innocent (people who tend to disagree battle it out in the comments). This question is deliberately designed to trick and confuse high school students in the exam. Now, I'm all for tricky problems, but this question isn't a problem: It's a question. And its certainly not a puzzle.

Puzzles are awesome. They confuse you, make your brain twist in new and unexpected ways, and a good puzzle will energize you. In fact, I love puzzles. Not just the jigsaws that we solve at home, but all kinds of puzzles. For instance, this is a classic one that I love.

You're at your house. You go one mile south, one mile east, and one mile north. You arrive back at your house. Where is your house?

This puzzle's great. It doesn't take much time to solve, and you'll probably be really happy with yourself after you've solved it (I'm not giving any spoilers) until I tell you that, in fact, there's more than one place on the earth where this can happen.

This is a kind of puzzle which I like to call a Fractal. Usually, fractals are these weird figures which pop up when we're talking about Chaos and stuff, which have the property that if you zoom in enough, you'll begin to see repeating patterns. I like them because there's more to them than meets the eye. You can keep going deeper and deeper.

Ooh, shiny.

And that's what the puzzle is! You can keep going deeper and deeper, and you can keep finding more and more places which fit the description. Here's another one just for fun.

Three gods are called, in no particular order, Eurus, Zephyr, and Aurorus. Eurus always speaks the truth, Zephyr always lies, while Aurorus can say whatever he wants which is really, truly random. You have to find out which god is which by asking each god exactly one question, which has to be a yes/no question. Just one problem: they won't answer you in English, they'll answer you in their language and will say either ja or da. You have no idea which is yes and which is no.

Harder than meets the eye. In fact, it's been called the hardest logic puzzle ever. You probably will not be able to solve it, like, ever.

Now let's shift from logic puzzles to math puzzles. Puzzles in math are largely the same- you have to try to find something. But in math, what you have to find is usually a proof- a set of statements which generalizes something.

The Ultimate Puzzlemaker

The person pictured up was a brilliant, awesome, mind blowing dude called David Hilbert. A German mathematician, he's one of the greats (He also had great aptitude at recognizing the other greats. Georg Cantor is practically known only because Hilbert found his ideas intriguing.). Hilbert was the leader of math during his age, and pretty much everything revolved around him.

Apart from that, he was also extremely chill. Not like Tesla, who tried to build a laser death ray, but like having-fun-on-the-beach kind of chill. We need more people like David Hilbert in the world.

What Hilbert is mostly known outside the sphere of math is his quote, one that's marked on his gravestone.

We Must Know. We Will Know.

 And if that doesn't sum up the greatness of this man, then this will. He was the classic puzzlemaker. In 1902, Hilbert gave a set of 23 problems which stumped the world, many of which remain unsolved today.

Hilbert's problems

In the International Congress of Mathematicians, 1900, Hilbert played his ace that would shock math for a very long time. By this time, Hilbert was already reputed, notably in invariance and the axioms of geometry. It was, perhaps, no surprise, considering where he came from.

Gottingen University

The University of Gottingen in Germany is perhaps the most influential university of all time. For a very long time, almost every new idea in math hailed from Gottingen. Apart from having a whole host of great minds like Gauss, Dirilecht, and Riemann, it would also become a significant location for one of the greatest physicists ever: Werner Heisenberg.

Hilbert's speech, delivered mostly in German outlined ten problems that were presented as goals for mathematicians in the coming years. Unlike the more recent Millennium Prize Problems, these looked forward to the future and acted as guidelines for further research. These were the puzzles that would shape modern math as we have it today.

The problems are from incredibly diverse fields, ranging from set theory (problem 1, the continuum hypothesis) to complex analysis (problem 8, the Riemann Hypothesis and other prime related problems). You can find a complete list of them here.

Further

That's the problems. Now, I'm planning to do a section where I discuss each of Hilbert's problems in some detail, but we'll see how well that goes considering it's going to take years just to understand them. Anyways, it'll be a lot of fun!

Thanks for reading!