We had some very good submissions showing us the ways that different pupils approached the activity and the way that they presented their results.

We were pleased to receive the following pdf from Angus at Canberra Grammar School in Australia. Here you see just a snippet so why not download the rest?

You can continue this chart forever, and it shows how every value can be made.

From Neve and Saba recorded their work in this way:

It would go on with the amount of money in the bags doubling (which is the pattern, we found out) forever!

We get our totals like this:

We added up the amounts of money in each of the bags to get the amount of money in the total column. 1p+2p+4p=7p

Rik from Cheongna Dalton School in South Korea, who was the youngest to send in their work, sent in this:

My mind put 1 penny in the first money bag. Then 2 pennies for the second money bag. But not 3 pennies for the third money bag. I put 4 pennies since I couldn't make four.

Then 4+1=5.

Then 4+2=6.

Then 4+2+1=7.

So I put 8 pennies in the last money bag.

Then 8+1=9.

Then 8+2=10.

Then 8+2+1=11.

Then 8+4=12.

Then 8+4+1=13.

Then 8+4+2=14.

Then if you add all of them together it equals 15.

Thank you to all who sent in their work its good to see the different ways of working and presenting.

We were pleased to receive the following pdf from Angus at Canberra Grammar School in Australia. Here you see just a snippet so why not download the rest?

See the rest of this exceptionally good presentation here.pdf

Laolu from the same school went to great lengths, as did Angus, to produce another presentation:

Laolu from the same school went to great lengths, as did Angus, to produce another presentation:

The full pdf can be found here.pdf

Bethany from Christ Church sent in the following explanation:

Step 1. Bag 1 has to have 1 penny in it to make 1 pence.

Step 2. To make 2 pence, bag two has to either have 1 penny or 2 pennies in it. However, if it has 1 penny in it, you can't make 3 pence. This would mean we would need yet another 1 penny to make 3 pence, so Bag 2 has to have two pennies in it.

Step 3. You can now make 1, 2 and 3 pence, but not four. This means that bag 3 has to have either 1, 2 or 4 pence inside it. Again, using a 1 pence or a two pence would mean you wouldn't be able to make higher values, so bag 3 has to have 4 pennies in it.

Step 4. You can now make all values up to 7 pence, so the next bag needs to hold a 1 pence, 2 pennies, 3 pennies, 4 pennies, 6 pennies or 7 pennies. however, (yet again!) this choice becomes easier when you think of the other values up to 15 pence. The only number of pennies you can have in this bag to ensure you could make all other digits is 8 pennies.

Solution: Bag 1 has 1 penny, bag 2 has 2 pennies, bag 3 has 4 pennies and bag 4 has 8 pennies.

Another way of proving this is using base two.

As you can see, we can use just the numbers in base two as the numbers of pennies, as I am going to prove below:

Bethany from Christ Church sent in the following explanation:

Step 1. Bag 1 has to have 1 penny in it to make 1 pence.

Step 2. To make 2 pence, bag two has to either have 1 penny or 2 pennies in it. However, if it has 1 penny in it, you can't make 3 pence. This would mean we would need yet another 1 penny to make 3 pence, so Bag 2 has to have two pennies in it.

Step 3. You can now make 1, 2 and 3 pence, but not four. This means that bag 3 has to have either 1, 2 or 4 pence inside it. Again, using a 1 pence or a two pence would mean you wouldn't be able to make higher values, so bag 3 has to have 4 pennies in it.

Step 4. You can now make all values up to 7 pence, so the next bag needs to hold a 1 pence, 2 pennies, 3 pennies, 4 pennies, 6 pennies or 7 pennies. however, (yet again!) this choice becomes easier when you think of the other values up to 15 pence. The only number of pennies you can have in this bag to ensure you could make all other digits is 8 pennies.

Solution: Bag 1 has 1 penny, bag 2 has 2 pennies, bag 3 has 4 pennies and bag 4 has 8 pennies.

Another way of proving this is using base two.

As you can see, we can use just the numbers in base two as the numbers of pennies, as I am going to prove below:

You can continue this chart forever, and it shows how every value can be made.

From Neve and Saba recorded their work in this way:

Number of bags

1

2

3

4

5

6

1

2

3

4

5

6

Amount of money

1p

2p

4p

8p

16p

32p

1p

2p

4p

8p

16p

32p

Total money you can get

1p

3p

7p

15p

31p

63p

1p

3p

7p

15p

31p

63p

It would go on with the amount of money in the bags doubling (which is the pattern, we found out) forever!

We get our totals like this:

Number of bags

1

2

3

1

2

3

Amount of money.

1p

2p

4p

1p

2p

4p

Total money you can get

1p

3p

7p

1p

3p

7p

We added up the amounts of money in each of the bags to get the amount of money in the total column. 1p+2p+4p=7p

Rik from Cheongna Dalton School in South Korea, who was the youngest to send in their work, sent in this:

My mind put 1 penny in the first money bag. Then 2 pennies for the second money bag. But not 3 pennies for the third money bag. I put 4 pennies since I couldn't make four.

Then 4+1=5.

Then 4+2=6.

Then 4+2+1=7.

So I put 8 pennies in the last money bag.

Then 8+1=9.

Then 8+2=10.

Then 8+2+1=11.

Then 8+4=12.

Then 8+4+1=13.

Then 8+4+2=14.

Then if you add all of them together it equals 15.

Thank you to all who sent in their work its good to see the different ways of working and presenting.