Pebbles

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?

Bracelets

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

Sweets in a Box

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

Six Ten Total

Age 7 to 11
Challenge Level

Take sixteen dice - six of one colour and ten of another.
Here we use blue and red.

Warm up: check the total of the faces showing is 84 when the dice are arranged as shown:

To play the Six Ten Total challenge:

all the blue need to have the same number on the top
all the red need to have the same number on the top
there always needs to be six of one colour and ten of the other
the red dice have to use a different number to the blue
there is always a difference of two between the numbers on the blue and red dice.

THE MAIN CHALLENGE

What are the possible arrangements when you choose your own numbers for the dice using the rules above?
What is the total for each of these arrangements?
What do you notice about your arrangements and the corresponding totals?
Explain what you notice. What else do you notice?

TAKING IT FURTHER

Can you prove any of the things you've noticed from the main challenge are always true?
Instead of having a difference of two between the numbers showing on each face of the blue and the red dice, choose a new difference.

Now, what totals can you find? What do you notice? Explain what you notice.

AND NOW

Can you predict - without having to make them - what numbers need to be on the faces that you can see on the blue and the red dice to make a total of 42?
Are there more ways to make a total of 42?
Explain your reasoning.

What happens if you change the number of blue and red dice yet keep the structure the same? It must be a square and both sets of dice must still make triangles.
Can you make 42?
Explain your reasoning.

This problem featured in a preliminary round of the Young Mathematicians' Award 2014.

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