Proof - June 2005, All Stages

Problems

problem icon

Ring a Ring of Numbers

Stage: 1 Challenge Level: Challenge Level:2 Challenge Level:2

Choose four of the numbers from 1 to 9 to put in the squares so that the differences between joined squares are odd.

problem icon

More Numbers in the Ring

Stage: 1 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?

problem icon

Number Differences

Stage: 2 Challenge Level: Challenge Level:1

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

problem icon

Crossings

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

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

problem icon

Diagonal Sums

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

In this 100 square, look at the green square which contains the numbers 2, 3, 12 and 13. What is the sum of the numbers that are diagonally opposite each other? What do you notice?

problem icon

Rabbit Run

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

Ahmed has some wooden planks to use for three sides of a rabbit run against the shed. What quadrilaterals would he be able to make with the planks of different lengths?

problem icon

Partitioning Revisited

Stage: 3 Challenge Level: Challenge Level:1

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

problem icon

Multiplication Square

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

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

problem icon

Cubes Within Cubes Revisited

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

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

problem icon

Pentagonal

Stage: 4 Challenge Level: Challenge Level:1

Can you prove that the sum of the distances of any point inside a square from its sides is always equal (half the perimeter)? Can you prove it to be true for a rectangle or a hexagon?

problem icon

Screen Shot

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

A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .

problem icon

Two Points Plus One Line

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

Draw a line (considered endless in both directions), put a point somewhere on each side of the line. Label these points A and B. Use a geometric construction to locate a point, P, on the line,. . . .

problem icon

Golden Construction

Stage: 5 Challenge Level: Challenge Level:1

Draw a square and an arc of a circle and construct the Golden rectangle. Find the value of the Golden Ratio.

problem icon

Golden Eggs

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

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.

problem icon

Golden Fractions

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

Find the link between a sequence of continued fractions and the ratio of succesive Fibonacci numbers.