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### Number and algebra

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# Gift of Gems

### Why do this problem?

This problem is a quirky puzzle that gives students an opportunity to express algebraically what is known in order to deduce relationships. It is a nice example of a problem that can be solved using algebra but also by finding a neat way to think about the situation.

### Possible approach

This problem could follow on from work on What's it Worth?

Once students have had a chance to think about the problem, take some time to discuss solution methods. One neat way to think about the problem is to realise that passing on three gems and receiving three different gems back leaves each jeweller with one of each gem plus their original collection minus 4...

### Key questions

What does each jeweller start with?

What does each jeweller end up with?

What can you deduce, knowing that each jeweller ends up with the same?

### Possible Extension

All-variables Sudoku gives a challenging workout with lots of opportunity to practise simultaneous equations.

### Possible Support

What's it Worth? gives students an opportunity to think strategically about how to work out the value of unknowns.## You may also like

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Age 14 to 16

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- Problem
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- Student Solutions
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This problem is a quirky puzzle that gives students an opportunity to express algebraically what is known in order to deduce relationships. It is a nice example of a problem that can be solved using algebra but also by finding a neat way to think about the situation.

This problem could follow on from work on What's it Worth?

Once students have had a chance to think about the problem, take some time to discuss solution methods. One neat way to think about the problem is to realise that passing on three gems and receiving three different gems back leaves each jeweller with one of each gem plus their original collection minus 4...

What does each jeweller start with?

What does each jeweller end up with?

What can you deduce, knowing that each jeweller ends up with the same?

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?