# Communicating and reflecting - September 2009, Stage 3&4

This month NRICH gets involved in the conversational side of mathematics, with problems which naturally lead to discussion and sharing of ideas and methods. Which forms of communication and presentation of the mathematics will prove to be the most effective? Through discussion and explanation you will refine your thinking and perhaps make connections with other areas of mathematics.

## Problems

### How Much Can We Spend?

##### Age 11 to 14 Challenge Level:

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

### Sticky Numbers

##### Age 11 to 14 Challenge Level:

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

### Where Can We Visit?

##### Age 11 to 14 Challenge Level:

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

### Chances Are

##### Age 14 to 16 Challenge Level:

Which of these games would you play to give yourself the best possible chance of winning a prize?

##### Age 14 to 16 Challenge Level:

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

### Carbon Footprints

##### Age 14 to 16 Challenge Level:

Is it really greener to go on the bus, or to buy local?

### Pythagoras Proofs

##### Age 14 to 16 Challenge Level:

Can you make sense of these three proofs of Pythagoras' Theorem?