# Freedom and Constraints

Mathematicians often explore structures, notice patterns, find rules and make generalisations. Once they feel they understand a structure, they then push at the boundaries, change the problem, and explore the new structures that emerge.

Here, we offer you some situations which may initially seem constrained. Take a look, there's more room to manoeuvre than you might initially imagine!

### Going to the Cinema

If every seat in the cinema is filled, and you know how much money was taken, can you work out who was in the audience? A chance to explore some simple examples of Diophantine Equations.

### Measuring with Rods

How many different lengths is it possible to measure with a set of three rods? Explore this problem, and make some discoveries about number bases.

### Climbing the Stairs

There are lots of ways to climb a staircase, taking one or two steps at a time. Explore this and related problems, and find out about some special number sequences.

### Combining Lengths

Start by simply putting rods together to measure different lengths. As you explore, you may make some discoveries that will lead you into the world of number theory.

### A Trip to the Cinema

There are plenty of opportunities to explore some mathematics on a trip to the cinema. From the cost of the tickets and the number of empty seats, to the price of the popcorn or buying some sweets, these problems will keep you busy while you're waiting for the trailers...

### Making Life Easier

Why do a task the hard way, when a little bit of thought can save you a lot of time? Here's a chance to find some efficient methods to solve some tricky problems.