this month, Stage 3 & 4

October 2001

 

Problems

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Appearing Square

Stage:3 Challenge Level:Challenge Level:1

Make an eight by eight square, the layout is the same as a chessboard. You can print out and use the square below. What is the area of the square? Divide the square in the way shown by the red dashed. . . .

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Thirty Nine, Seventy Five

Stage:3 Challenge Level:Challenge Level:1

We have exactly 100 coins. There are five different values of coins. We have decided to buy a piece of computer software for 39.75. We have the correct money, not a penny more, not a penny less! Can. . . .

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Serendipity

Stage:3 Challenge Level:Challenge Level:1

As is often the case when looking for something, you become distracted by other more exciting ideas. Such was the case recently when retrieving some old tiling patterns. How could I have overlooked. . . .

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Rhombicubocts

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

Each of these solids is made up with 3 squares and a triangle around each vertex. Each has a total of 18 square faces and 8 faces that are equilateral triangles. How many faces, edges and vertices. . . .

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Aba

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

In the following sum the letters A, B, C, D, E and F stand for six distinct digits. Find all the ways of replacing the letters with digits so that the arithmetic is correct.

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Framed

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

Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . .

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Disappearing Square

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

Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .

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Unit Fractions

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

Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation.

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Common Divisor

Stage:4 Challenge Level:Challenge Level:1

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

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Converse

Stage:4 Challenge Level:Challenge Level:1

Clearly if a, b and c are the lengths of the sides of a triangle and the triangle is equilateral then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true, and if so can you prove it? That is if. . . .

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Fitting In

Stage:4 Challenge Level:Challenge Level:1

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . .

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Cubics

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

Knowing two of the equations find the equations of the 12 graphs of cubic functions making this pattern.

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Pent

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

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.