Generalising - May 2004, Stage 4&5

Problems

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Pair Products

Stage: 4 Challenge Level: Challenge Level:1

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

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Three Frogs

Stage: 4 Challenge Level: Challenge Level:1

Three frogs hopped onto the table. A red frog on the left a green in the middle and a blue frog on the right. Then frogs started jumping randomly over any adjacent frog. Is it possible for them to. . . .

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Perfectly Square

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

The sums of the squares of three related numbers is also a perfect square - can you explain why?

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Squirty

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Using a ruler, pencil and compasses only, it is possible to construct a square inside any triangle so that all four vertices touch the sides of the triangle.

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Generally Geometric

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

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

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Golden Fibs

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

When is a Fibonacci sequence also a geometric sequence? When the ratio of successive terms is the golden ratio!

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Water Pistols

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

With n people anywhere in a field each shoots a water pistol at the nearest person. In general who gets wet? What difference does it make if n is odd or even?