Representation - October 2008, All Stages

Problems

Magic Plant

Stage: 1 Challenge Level:

On Friday the magic plant was only 2 centimetres tall. Every day it doubled its height. How tall was it on Monday?

Find the Difference

Stage: 1 Challenge Level:

Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.

Biscuit Decorations

Stage: 1 and 2 Challenge Level:

Andrew decorated 20 biscuits to take to a party. He lined them up and put icing on every second biscuit and different decorations on other biscuits. How many biscuits weren't decorated?

Stage: 1 and 2 Challenge Level:

Place six toy ladybirds into the box so that there are two ladybirds in every column and every row.

Three Squares

Stage: 1 and 2 Challenge Level:

What is the greatest number of squares you can make by overlapping three squares?

Five Coins

Stage: 2 Challenge Level:

Ben has five coins in his pocket. How much money might he have?

Stage: 2 Challenge Level:

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

Crossed Ends

Stage: 3 Challenge Level:

Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?

Legs Eleven

Stage: 3 Challenge Level:

Take any four digit number. Move the first digit to the 'back of the queue' and move the rest along. Now add your two numbers. What properties do your answers always have?

Domino Square

Stage: 2, 3 and 4 Challenge Level:

Use the 'double-3 down' dominoes to make a square so that each side has eight dots.

Differences

Stage: 3 Challenge Level:

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?

Tet-trouble

Stage: 4 Challenge Level:

Show that is it impossible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units...

Hexy-metry

Stage: 4 Challenge Level:

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

Napoleon's Theorem

Stage: 4 and 5 Challenge Level:

Triangle ABC has equilateral triangles drawn on its edges. Points P, Q and R are the centres of the equilateral triangles. What can you prove about the triangle PQR?

Sixty-seven Squared

Stage: 5 Challenge Level:

Evaluate these powers of 67. What do you notice? Can you convince someone what the answer would be to (a million sixes followed by a 7) squared?

Chord

Stage: 5 Challenge Level:

Equal touching circles have centres on a line. From a point of this line on a circle, a tangent is drawn to the farthest circle. Find the lengths of chords where the line cuts the other circles.

Pythagoras for a Tetrahedron

Stage: 5 Challenge Level:

In a right-angled tetrahedron prove that the sum of the squares of the areas of the 3 faces in mutually perpendicular planes equals the square of the area of the sloping face. A generalisation. . . .