Small and Beautiful - September 2003, All Stages

Problems

Ladybirds in the Garden

Stage: 1 Challenge Level:

In Sam and Jill's garden there are two sorts of ladybirds with 7 spots or 4 spots. What numbers of total spots can you make?

A Conversation Piece

Stage: 2 Challenge Level:

Take the number 6 469 693 230 and divide it by the first ten prime numbers and you'll find the most beautiful, most magic of all numbers. What is it?

Which Is Quicker?

Stage: 2 Challenge Level:

Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why?

Folding, Cutting and Punching

Stage: 2 Challenge Level:

Exploring and predicting folding, cutting and punching holes and making spirals.

Shapely Tiling

Stage: 2 Challenge Level:

Use the interactivity to make this Islamic star and cross design. Can you produce a tessellation of regular octagons with two different types of triangle?

1, 2, 3, 4, 5

Stage: 2 Challenge Level:

Using the numbers 1, 2, 3, 4 and 5 once and only once, and the operations x and ÷ once and only once, what is the smallest whole number you can make?

Sept03 Sept03 Sept03

Stage: 2 Challenge Level:

This number has 903 digits. What is the sum of all 903 digits?

Folding

Stage: 2 Challenge Level:

What shapes can you make by folding an A4 piece of paper?

Smaller and Smaller

Stage: 2 Challenge Level:

Can you predict, without drawing, what the perimeter of the next shape in this pattern will be if we continue drawing them in the same way?

Fitted

Stage: 2 Challenge Level:

Nine squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18 cm can be fitted together to form a rectangle. What are the dimensions of the rectangle?

LOGO Challenge - the Humble Square

Stage: 2, 3 and 4 Challenge Level:

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.

Sept 03

Stage: 3 Challenge Level:

What is the last digit of the number 1 / 5^903 ?

Repetitiously

Stage: 3 Challenge Level:

The number 2.525252525252.... can be written as a fraction. What is the sum of the denominator and numerator?

Hidden Squares

Stage: 3 Challenge Level:

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

Adding Triangles

Stage: 3 Challenge Level:

What is the total area of the first two triangles as a fraction of the original A4 rectangle? What is the total area of the first three triangles as a fraction of the original A4 rectangle? If. . . .

Terminology

Stage: 3 Challenge Level:

Triangle ABC is isosceles while triangle DEF is equilateral. Find one angle in terms of the other two.

Rati-o

Stage: 3 Challenge Level:

Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle?

Three Times Seven

Stage: 3 Challenge Level:

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

Six Times Five

Stage: 3 Challenge Level:

How many six digit numbers are there which DO NOT contain a 5?

Excel Investigation: Power Crazy

Stage: 3 and 4 Challenge Level:

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

Tiny Nines

Stage: 4 Challenge Level:

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.

Dodecawhat

Stage: 4 Challenge Level:

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.

Semi-detached

Stage: 4 Challenge Level:

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.

Fac-finding

Stage: 4 Challenge Level:

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.

Circumnavigation

Stage: 4 Challenge Level:

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

Ruler

Stage: 5 Challenge Level:

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?

Small Steps

Stage: 5 Challenge Level:

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

Small and Beautiful - September 2003, All Stages

Problems

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Stage: 2, 3 and 4 Challenge Level:

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Stage: 3 and 4 Challenge Level:

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