Whether you want to understand viruses, win at lotto or solve a rubik's cube – you'll need some help from combinatorics.
Euler may not have cracked this problem completely, but it led to a lot of important work, including on what we today know as sudoku.
How many possible genetic relationships are there between a collection of different species? The answer is mind-bogglingly large.
Given there's a finite number of notes on a scale, can we still find a brand new melody? Perhaps they've all been written already!
Can we always find order in systems that are disordered? If so, just how large does a system have to be to contain a certain amount of order?
How do you best allocate students to universities, doctors to hospitals, or kidneys to transplant patients? It's a tough problem that has earned this year's Memorial Prize in Economics.
Andy Murray and Laura Robson made a good team at London 2012, bringing home silver in the mixed doubles. But how do you make sure that the competing pair is the best you can pick from the team?
Water is essential for life on Earth, and it is a resource we all take for granted. Yet it has many surprising properties that have baffled scientists for centuries. Seemingly simple ideas such as how water freezes are not understood because of water's unique properties. Now scientists are utilising increased computer power and novel algorithms to accurately simulate the properties of water on the nanoscale, allowing complex structures of hundreds or thousands of molecules to be seen and understood.