
Princess Catharina's Gold Rings
Years 4  10

Preparation
 Fold and tear a piece of paper into twelve (12) pieces.
 Write one of these numbers on each piece:
5, 6, 7, 8, 9, 10, 11, 14, 15, 16, 17 and 40.
 Write the title of this challenge and today's date on a fresh page in your maths journal.
The Story
 Princess Catharina has twelve gold rings.
 Each ring is in its own box.
 Inside each box is a piece of paper with the number of grams of gold used to make the ring.
 The ring boxes are kept in the top drawer, the middle drawer and the bottom drawer.
The Princess loved maths at school.
She even studied maths when she went to university.
So she made up a puzzle for the servant in charge of the rings.

Princess Catharina has also made a video,
Princess Catharina's Trinket Chest,
to help you understand fractions.

(Based on free clip art from Stockio.com) 
She said:
You can put any ring in any of the three drawers.
BUT
 The top drawer must have three (3) rings.
 The middle drawer must have four (4) rings.
 The bottom drawer must have five (5) rings.
AND
 The weight of gold in the middle drawer must be twice the weight of gold in the top drawer.
Which rings might have been in the bottom drawer?

Getting Started
When a mathematician sees this problem for the first time, they don't know any more than you do.
But they do know how to work like a mathematician to try to find an answer.
Click the link to open their method of finding an answer.
You can read it on screen or print it.
Copy this sentence into your journal and use the first dot point on Working Mathematically to finish it,
When a mathematician starts a problem they ...
So all you have to do is play!
Just muck around with the numbers until they do what the Princess wants.
Suggestion
If you look further down the Working Mathematically page you will see that one strategy a mathematician uses is 'make a model'.
Here's one way you could do that.
 Fold a piece of paper into three parts to represent the three drawers.
 Mark them Top (3), Middle (4) and Bottom (5) so you remember one of the rules.
Challenge 1
 Start playing around with your twelve pieces of paper until you find a way that works.
 Keep trying for about ten (10) minutes.
 If you find an answer, draw a picture of the three drawers in your journal.
Show which rings go in each drawer and the total grams of gold in each drawer.
If you don't find an answer after ten minutes, go to the bottom of the page and use the answer there.
Have fun exploring Princess Catharina's Gold Rings.
Challenge 2
There are at least eleven (11) ways to put all the rings in drawers with 5 and 6 in the top drawer.
 Find two (2) of these ways.
 Record your answers in your journal.
 Draw a quick cartoon of your face.
Give it a thought bubble to explain a good idea that helped you find these answers.
You can add more thought bubbles if you want to.
If a problem is a good one, a mathematician can't do it all in one go.
So they keep good journal notes and diagrams about what they know so far.
Then they take a break.
You can too.
You don't have to do all this problem today.
 But there is more you could do.
 So make good journal notes about your best thinking  and the things that didn't work.
 Then come back another time.


Challenge 3
So far you have found three (3) ways to put the rings in the drawers.
At least two of them have 5 and 6 in the top drawer.
You have been told there are at least 11 ways to do it with 5 and 6 in the top drawer.
 Find all the ways to put the rings in the drawers with 5 and 6 in the top.
This page will help you if you need it.
Note:
 The help page starts you off with finding [5, 6, __] solutions but doesn't help you find all of them.
 There is one more with [5, 6, 14] and you will also need to try 5 and 6 with 15, 16, 17 and 40.
Challenge 4
Princess Catharina told her servant that she likes her problem because she thinks it is possible for every ring to be in the bottom drawer at least once.
 The servant decided to check. Was the Princess right?
Challenge 5
 How many solutions are there altogether for the Princess's Gold Ring problem?
 How do you know when you have found them all?


A Side Order of Problems
 The servant got sick of the rings never being the same way and asked the Princess if she could think of one more rule so that there could only be one answer.
 Keeping the other rules, what new rule could the Princess add?
 One (1) gram is about the weight of a paper clip or a raisin.
 Estimate how much 1 gram of gold is worth today. Record your estimate.
 Check your estimate by using your browser to search 'gold price per gram Australia'.
Record the result.
How close were you?
 Use the search result to calculate the value of the gold used to make the Princess's rings.
 The Princess decided to use the 3 drawers for something else.
She asked the servant to move the rings to the two drawer side and make up a puzzle for her.
 Pretend you are the servant.
Create your own two drawer puzzle for the twelve rings.


Just Before You Finish
 A mathematician must be able to explain their discoveries to other mathematicians, so when you have finished finding out stuff, prepare a report of your discoveries. You might write a report, or make a slide show, or a video, or a poster or explain in some other way.
 Read your Working Like A Mathematician page again and write three or more sentences explaining how you worked like a mathematician.
Answers & Discussion
TOP (3)
5 ... 6 ... 9
Total = 20

MIDDLE (4)
7 ... 8 ... __ ... 15
Total = 40

BOTTOM (5)
__ ... __ ... __ ... __ ... __
Total = __


Help for Challenge 1
If you couldn't find a solution to the first challenge after 10 minutes, you can use this one.
Make it on your folded paper, check that it obeys the rules and finish it.
Then copy it into your journal with all the answers.
These notes were originally written for teachers. We have included them to support parents to help their child learn from Princess Catharina's Gold Rings. In these notes it is a different story shell and the problem is called Eureka.

