# combinatorics

 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. Many mathematicians find the pure and tight patterns of juggling as irresistible as those of mathematics. Burkard Polster explains how to get to grips with the bewildering range of juggling possibilities and invites you to do your own virtual juggling. What's your strategy for love? Hold out for The One, or try and avoid the bad ones? How long should you wait before cutting your losses and settling down with whoever comes along next? John Billingham investigates and saves the national grid in the process. Bonuses are a fact of business life. Last year the Guardian newspaper calculated that the cash rewards paid to London's financial chiefs comfortably outstripped the UK's entire transport budget. With such large sums at stake, envy is bound to raise its ugly head, nver a good thing for company morale. So how should you decide who gets how much? Steven J. Brams suggests a method that's not only fair, but also encourages honesty. Computer scientists prove how long it should take you to solve Rubik's cube There are many different types of lottery around the world, but they all share a common aim: to make money. John Haigh explains why lotteries are the way they are. In the early days of the UK National Lottery, it was quite common to see newspaper articles that looked back on what numbers had recently been drawn, and attempted to identify certain numbers as "due" or "hot". Few such articles appear now, and John Haigh thinks that perhaps the publicity surrounding the lottery has enhanced the nation's numeracy.