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Money Bags

Stage: 2 Challenge Level: Challenge Level:2 Challenge Level:2
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?

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:

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:

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
Amount of money
Total money you can get


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
Amount of money. 
Total money you can get

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.

Year 4 at Abbey Primary School sent in their solution as a picture that shows all the combinations, well done!

Thank you to all who sent in their solutions.  It's good to see the different ways of working and presenting.