Russia

Mathematicians

One of the most famous unsolved problems relating to number theory in mathematics comes from a native of Russia. Goldbach (1690 - 1764) was a professor of mathematics in St. Petersburg. He enjoyed a good relationship with his fellow mathematicians at the time and traveled all over Europe meeting them. In particular, Goldbach wrote to Euler on a number of occasions discussing number theory problems.

In 1742, in a letter to Euler, Goldbach wrote that every even integer greater than 2 can be represented as the sum of two primes. He also conjectured that every odd number is the sum of three primes. A prime number is a number greater than 1 with only 2 factors; 1 and itself. This problem seems simple enough, doesn't it? Some simple examples are 4=2+2, 6=3+3, 8=3+5, 100=53+47, and so on. In fact it is possible to find hundreds of examples where this works. All it would take is to find one example where this is *not* true to disprove this famous math problem that has been outstanding for well over 200 years. A fellow Russian proved in 1937 that every odd number from some point onwards can be written as the sum of 3 primes which has been called Goldbach's ‘soft' conjecture but still could not prove that that every even integer greater than 2 can be represented as the sum of two primes.

This problem is so famous that in 2000 a publisher of a book on Goldbach's theorem, Faber and Faber offered a prize of $1,000,000 to anyone who could produce a proof before April 2002. No one stepped forward to claim this prize. If you could prove this conjecture – or alternatively and more easily, perhaps, find just one example that contradicts it, then fame and fortune will await you. Most importantly, it shows that there is still much about math we do not know for certain, even though we have been studying it for thousands of years.

Problem

Markov is another famous mathematician because he helped develop probability theory. Some of his theory is used to attempt to make predictions in such non predictable areas such as weather forecasting and the stock market. He is also well known for his random walk theory.

Here an illustration of the theory. Walking home one night, you see a man staggering out of a pub. He is trying to walk straight, but he is unfortunately under the influence of too much vodka [imagine you are in Russia] and so is stumbling a bit. First he takes a step to the right or maybe two and then he takes a step to the left to try and correct. He eventually makes it home even though he does not take a step in a straight forward direction. We can simulate the “Drunkards Walk” with grid paper and a coin.

At every random coin toss we take a step either diagonally left [HEADS] or diagonally right [TAILS]. Do we eventually end up at home? If not, what can we do to make sure we do end up at home eventually? How many tries does it take you to end up home?

Thinking about the probability behind tossing the coin, if it is ½ chance of getting a head and a ½ chance of getting a tail, why do you not always end up in a straight line from the bar at home?

Here is an example where we toss H, H, T, H, T, T, T, H, T, T (see image below):

Sphere Problem - Cutting a Sphere

As you know, the earth has an equator that cuts it into a northern and southern hemisphere. It also has a prime meridian that cuts it into east and west hemispheres. Now imagine you could divide the earth with 4 straight cuts. You are probably thinking “ah that would divide the earth into 8 regions!”. But did you know you can divide the world into fifteen different regions with just 4 cuts? How could you do it?

Try solving the International Color Challenge!