### Helen's Conjecture

Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?

### Marbles

I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?

### More Marbles

I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?

# How Much Can We Spend?

##### Stage: 3 Challenge Level:

A country has decided to have just two different coins.

It has been suggested that these should be 3z and 5z coins.

The shops think this is a good idea since most totals can be made.

 $2\times3z+1\times 5z=11z$ $7 \times 3z + 2 \times 5z = 31z$

Unfortunately some totals can't be made, for example 4z.

Is there a largest total that cannot be made?

How do you know?

They have decided that they will definitely have 3z coins but can't make up their minds about the other coin.

Experiment with other pairings containing 3z, and explore which totals can be made.

Can you find a relationship between 3z, the second coin, and the totals that can and can't be made?

In other countries they have also decided to have just two coins, but instead of the 3z coins they have chosen a different prime number.

Can you find a relationship between pairs of coin values and the totals that can and can't be made with them?

NOTES AND BACKGROUND

The coin problem (also referred to as Frobenius coin problem or Frobenius problem) is a mathematics problem associated with the German mathematician Ferdinand Georg Frobenius and often introduced in the context of making exact change given the availability of coins of specific denominations. To read about it go to Wikipedia.