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Do Unto Caesar

At the beginning of the night three poker players; Alan, Bernie and Craig had money in the ratios 7 : 6 : 5. At the end of the night the ratio was 6 : 5 : 4. One of them won $1 200. What were the assets of the players at the beginning of the evening?

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Plutarch's Boxes

According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have their surface areas equal to their volumes?

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Using some or all of the operations of addition, subtraction, multiplication and division and using the digits 3, 3, 8 and 8 each once and only once make an expression equal to 24.

Diminishing Returns

Age 11 to 14 Challenge Level:

Why do this problem

The problem offers opportunities to think about area, proportion and fractions, while offering an informal introduction to the mathematics of infinity and convergence which would not normally be met by younger students, to tempt their curiosity.

Possible approach

Show the first image from the problem for a short while and invite students to look at it in silence. Then hide the image and give them some time to reflect on what they saw and then discuss with a partner, and finally share with the whole class.

"What did you see?"
"Did anyone see it differently?"

Once students have shared their thoughts, show them the image again to see how their impressions of it compared with the actual picture. Then pose the following question:

"What questions might a mathematician want to explore?"
Give them some time to come up with some suggestions. If they are struggling to think of any, there are some suggestions in the "Key questions" section below.

Challenge students to work out the proportion of the square that is shaded blue at each stage. There are three more images in the problem that you could use in the same way - invite students to create their own versions, and to work out the proportion taken up by each size of triangle. This could make a lovely wall display using colourful paper and explanations of the reasoning used to work out the relationships between different triangles.

You might like to finish off with a discussion about what would happen if the patterns carried on forever. This is explored some more in the problem Vanishing Point.

Key questions

  • For the first stage of the pattern, how many blue triangles are there? What about at the second... third... fourth stage?
  • How do the areas of each blue triangle compare with the next size up?
  • How could I create the image from coloured paper?
  • What proportion of the image is shaded blue?
  • What if it carried on forever?

Possible extension

Vanishing Point uses the same starting point and goes on to explore the question of continuing the sequence indefinitely.

Possible support

In the Teachers' Resources to Inside Seven Squares there is a demonstration of how paper folding could be used to help students to create the shapes, and see the relationship between areas.