Small and Beautiful - September 2003, Stage 4&5

Problems

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LOGO Challenge - the Humble Square

Stage: 2, 3 and 4 Challenge Level: Challenge Level:1

Look at how the pattern is built up - in that way you will know how to break the final pattern down into more manageable pieces.

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Excel Investigation: Power Crazy

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

When is 7^n + 3^n a multiple of 10? Use Excel to investigate, and try to explain what you find out.

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Tiny Nines

Stage: 4 Challenge Level: Challenge Level:1

Find the decimal equivalents of the fractions one ninth, one ninety ninth, one nine hundred and ninety ninth etc. Explain the pattern you get and generalise.

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Dodecawhat

Stage: 4 Challenge Level: Challenge Level:1

Follow instructions to fold sheets of A4 paper into pentagons and assemble them to form a dodecahedron. Calculate the error in the angle of the not perfectly regular pentagons you make.

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Semi-detached

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.

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Fac-finding

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Lyndon chose this as one of his favourite problems. It is accessible but needs some careful analysis of what is included and what is not. A systematic approach is really helpful.

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Circumnavigation

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.

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Ruler

Stage: 5 Challenge Level: Challenge Level:1

The interval 0 - 1 is marked into halves, quarters, eighths ... etc. Vertical lines are drawn at these points, heights depending on positions. What happens as this process goes on indefinitely?

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Small Steps

Stage: 5 Challenge Level: Challenge Level:1

Two problems about infinite processes where smaller and smaller steps are taken and you have to discover what happens in the limit.