this month, Stage 3 & 4

June 2000

 

Problems

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LOGO Challenge - Tiling with Equilateral Triangles

Stage:3 Challenge Level:Challenge Level:1

At a maths conference someone said that - "not much can be done when tiling the plane with equilateral triangles." Below are just four tilings that grabbed my attention. Try replicating them in LOGO. . . .

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Ones Only

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

Find the smallest whole number which, when mutiplied by 7, gives a product consisting entirely of ones.

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Please Explain

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

Visitors to Earth from the distant planet of Zub-Zorna were amazed when they found out that when the digits in this multiplication were reversed, the answer was the same! Find a way to explain. . . .

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Do Unto Caesar

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

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. . . .

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One Basket or Group Photo

Stage:2, 3, 4 and 5 Challenge Level:Challenge Level:3Challenge Level:3Challenge Level:3

Libby Jared helped to set up NRICH and this is her choice for our 10th anniversary. It's a problem suitable for a wide age range and best tackled practically.

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Tangentially Yours

Stage:4 Challenge Level:Challenge Level:1

Three circles are drawn tangentially to each other, their centres collinear, with two circles inside the third circle. The line joing A and B on the biggesr circle is tangential to the two smaller. . . .

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Zig Zag

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

Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?

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Cyclic Quad Jigsaw

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

A quadrilateral is put together by juxtaposing five smaller quadrilaterals as shown. Prove that if each of the five quadrilaterals is cyclic, then so is the resulting quadrilateral.

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Good Approximations

Stage:4 and 5 Challenge Level:Challenge Level:1

Solve quadratic equations and use continued fractions to find rational approximations to irrational numbers.