How hard is it to grasp open problems in mathematics? Do you need to be an adult, a genius or a research mathematician to enjoy pondering such problems? Or can a curious young person be an investigator as well?
By an “open problem”, I mean a problem in mathematics that humanity has not yet solved. This stands in contrast to typical problems that students get in exams and homework assignments. Those almost always have known solutions. While solving such "textbook" problems may be fun, open problems are a completely different ball game. Solve one of those and enter the books of history!
There are tens of thousands of open mathematical problems touched by humans and still unsolved. In fact, as you are reading this, thousands of research mathematicians and enthusiasts around the globe are thinking about open problems. Some are loosing sleep over it too. To solve one, you need endurance, creativity, and sometimes a touch of genius. Nevertheless, you don't need to be a genius to enjoy mathematics. I'm talking from experience here.
For some famous open problems in mathematics, getting an understanding of the meaning of the problem is a formidable task in its own right. It requires knowledge of the definitions and terms that setup the problem and make it interesting. Sometimes to appreciate such problems, one needs to have solved similar simpler problems earlier.
For example, one of the greatest open problems of these days is the 160 year old Riemann Hypothesis. Have a glance for example at this nice numberphile video featuring Edward Frenkel:
You probably follow the great exposition of Edward Frenkel just fine. However as the exposition continues, you may appreciate the fact that really understanding the details of the Riemann Hypothesis requires quite a bit of knowledge. This is at a level comparable to several university level mathematics courses.
University mathematics is great fun, but does one need to spend all this time learning mathematics before fully understanding famous unsolved problems? Certainly not! As I was recently reminded by Sunil Singh, author of Pi of Life, not all problems require so much mastery. We may actually entertain our children’s curiosity with a very famous problem called Collatz Conjecture:
Sunil posted a snippet of a conversation with with his young daughter where she calculated the sequence:
7, 22, 11, 34, 17, 52, 26, 13, 40, 20 10, 5, 16, 8, 4, 2, 1.
This may seem a bit arbitrary, but let's make some sense of it. Notice that every even number is followed by half it’s value. For example 22 is followed by 11 and 20 is followed by 10. Also notice that the sequence ends in the number 1.
Perhaps pause now to see what else you can observe about this sequence. What do you see?
Observe for example that after every odd number there is an even number.
After 7 we get 22.
After 11 we get 34.
After 17 we get 52.
After 13 we get 40.
After 5 we get 16.
Can you suggest some relationship between an odd number and the number that follows it? How about multiply it by 3 and then add 1? For example, 3 * 7 + 1 = 22. Try it for the other odd numbers as well. Note that the order of operation matters here. First multiply by 3, and only after that add 1.
So this sequence of numbers acts as follows: You start with some number (in our example above 7), and then repeat the following step:
If the current number is even take half of it. If your current number is odd, triple it and add 1.
This sequence of numbers is sometimes called the hailstone sequence because the numbers in the sequence are subject to both ascents (when odd) and descents (when even). The analogy is to hailstones in a cloud. Apparently hailstones sometimes fly up and down inside a cloud. But do they all fall down? Or do some fly forever?
Returning from the hailstone analogy to an open problem in mathematics we have the Collatz Conjecture:
Conjecture: For any starting value, a hailstone sequence eventually reaches the value 1.
In mathematics, a conjecture is a belief of something that is true, that has not yet been proven. Hence it may be that the conjecture is false. Proving that the conjecture is true or false is an open problem.
Researchers have tried to prove the Collatz Conjecture using a variety of methods and others have tried to disprove it by using computers and searching for huge numbers that end up in a cycle and never reach 1. But alas, all huge numbers tested reached one and to date, all proof attempts have failed. Indeed, it is an open problem.
So as Sunil did with his daughter I entertained my children with the hailstone sequence. To make things fun we played a little game. One of us chose a starting number and we took turns in saying out consecutive numbers in the sequence. The winner is the person that ends up with 1. For example, with three people, each marked with a different color you can get the following sequence:
6, 3, 10, 5, 16, 8, 4, 2, 1.
Here the red person decided to start with 6 and the green person won because she ended with 1.
We enjoyed playing a few rounds of this game. My daughters got the hang of the problem as we played and in the process had some arithmetic practice. It was common to want to start with 100 as the initial number. This yields 100, 50, 25, 76, 38, 19, 58, 29, 88, 44, 22,... and now we merged back with the sequence of Sunil's daughter.
Be careful for example with 27. Start with this value and your hailstone will fly high and take a long while until reaching the ground. How long? Maybe do it by hand, or work with your child using Excel for doing it (you may need to use "if" and "mod" or similar Excel functions).
Play around with your children for a while and see if they pose the famous open problem (Collatz conjecture) by themselfs:
Child: "Do you always end up in 1?"
Educator: "Dear child, nobody knows yet. People have tried starting the sequence with many numbers and all numbers tested reached 1. But nobody proved that it is true for all numbers. Maybe you can become famous forever by finding an initial (huge) hailstone value, that never hits the ground. Or maybe you can investigate the Collatz conjecture a bit further and one day help find a proof. You never know."