this month, Stage 3 & 4

August 1998

 

Problems

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Pebbles

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

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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Not a Polite Question

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

When asked how old she was, the teacher replied: My age in years is not prime but odd and when reversed and added to my age you have a perfect square...

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The Lady or the Lions

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

The King showed the Princess a map of the maze and the Princess was allowed to decide which room she would wait in. She was not allowed to send a copy to her lover who would have to guess which path. . . .

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On Time

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

On a clock the three hands - the second, minute and hour hands - are on the same axis. How often in a 24 hour day will the second hand be parallel to either of the two other hands?

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Square Routes

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

How many four digit square numbers are composed of even numerals? What four digit square numbers can be reversed and become the square of another number?

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Bi-cyclics

Stage:4 Challenge Level:Challenge Level:1

Two circles intersect at A and B. Points C and D move round one circle. CA and DB cut the other circle at E and F. What do you notice about the line segments CD and EF?

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Same Height

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

A trapezium is divided into four triangles by its diagonals. Suppose the two triangles containing the parallel sides have areas a and b, what is the area of the trapezium?

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Max Box

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

Three rods of different lengths form three sides of an enclosure with right angles between them. What arrangement maximises the area

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Obviously?

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

Find the values of n for which 1^n + 8^n - 3^n - 6^n is divisible by 6.

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Voting Paradox

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

Some relationships are transitive, such as `if A>B and B>C then it follows that A>C', but some are not. In a voting system, if A beats B and B beats C should we expect A to beat C?