August 2002, All Stages

Problems

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Domino Number Patterns

Stage: 1 Challenge Level: Challenge Level:2 Challenge Level:2

Can you work out the domino pieces which would go in the middle in each case to complete the pattern of these eight sets of 3 dominoes?

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Dining Ducks

Stage: 2 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Use the information about the ducks on a particular farm to find out which of the statements about them must be true.

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Special Numbers

Stage: 3 Challenge Level: Challenge Level:1

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

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And So on and So On

Stage: 3 Challenge Level: Challenge Level:1

If you wrote all the possible four digit numbers made by using each of the digits 2, 4, 5, 7 once, what would they add up to?

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Darts and Kites

Stage: 4 Challenge Level: Challenge Level:1

A rhombus PQRS has an angle of 72 degrees. OQ = OR = OS = 1 unit. Find all the angles, show that POR is a straight line and that the side of the rhombus is equal to the Golden Ratio.

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Archimedes and Numerical Roots

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

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

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Card Game (a Simple Version of Clock Patience)

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

Four cards are shuffled and placed into two piles of two. Starting with the first pile of cards - turn a card over... You win if all your cards end up in the trays before you run out of cards in. . . .

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Retracircles

Stage: 5 Challenge Level: Challenge Level:1

Four circles all touch each other and a circumscribing circle. Find the ratios of the radii and prove that joining 3 centres gives a 3-4-5 triangle.

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Pair Squares

Stage: 5 Challenge Level: Challenge Level:1

The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.

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Square Pair Circles

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.