50 Mathematical Ideas You Really Need to Know

case is to minimize the cost of transportation. In some linear programming problems the objective is to maximize (like maximizin ...

46 The travelling salesperson James Cook, based in Bismarck (North Dakota, USA), is a super salesperson for the Electra company, ...

785 miles from Chicago, and is nearer than the other options. Once in Dallas he has notched up 706 + 785 miles. He then has to c ...

distance he wants to save, but time. He draws up a new table that gives the travel time between the different cities in his terr ...

problem. Exact methods have been given which apply to 5000 cities or less and one has even successfully dealt with a particular ...

verified in polynomial time. The travelling salesperson problem is definitely one of these because checking whether a given rout ...

47 Game theory Some said Johnny was the smartest person alive. John von Neumann was a child prodigy who became a legend in the m ...

sign, as in the payoff −3, is that ATV will lose an audience of 3 million. The + payoffs are good for ATV and the − payoffs are ...

As before, the safe strategy for ATV is to choose the row which maximizes the worst that can happen. The maximum from {+1, −1, − ...

count of three: paper draws with paper, is defeated by scissors (since scissors can cut paper), but defeats stone (because it ca ...

and not confess – this would be the ‘win–win’ situation. the condensed idea Win–win mathematics ...

48 Relativity When an object moves, its motion is measured relative to other objects. If we drive along a major road at 70 miles ...

set of coordinates ( ,). It is also possible to synchronize these two coordinate systems. When the train passes the mark on the ...

The Lorentz factor The special theory of relativity Lorentz set out the mathematical equations which governed the connection bet ...

simple principles of relativity and the whole theory unfolded. In particular, he showed that the energy of a particle E is deter ...

Einstein’s four-dimensional space–time geometry (or Riemannian geometry) is curved. It displaces the Newtonian force of gravity ...

49 Fermat’s last theorem We can add two square numbers together to make a third square. For instance, 5^2 + 122 = 13^2. But can ...

make a solution of the equation. We can also see that some values of x, y and z are not solutions of the equation. For example x ...

values x = 3, y = 4 and z = 5 turn out to be a solution of the kind we are looking for because 3^2 + 4^2 = 9 + 16 = 5^2. From th ...

It looks like a small step to go from x^2 + y^2 = z^2 to x3 + y^3 = z^3. So, following the idea of reassembling one square aroun ...