AIMSSEC Puzzles

These 'trisquares' are made up of three squares and each has an area of 3 square units.
Can you fit them together to make an enlargement of the shape? What is its area?
Can you fit trisquares together to make enlargements of scale factors 3, 4 and 5? What are their areas?
Is it possible to make enlargements of all sizes by fitting trisquares together?
Squared paper would be useful for working on this challenge.

Imagine a large cube made up from 27 small red cubes.
Imagine dipping the large cube into a pot of yellow paint so the whole outer surface is covered, and then breaking the cube up into 27 small cubes.
How many of the small cubes will have yellow paint on their faces?
Will they all look the same?
How many red faces and how many yellow faces do they have?
Now imagine doing the same with other big cubes made up from small red cubes.
What can you say about the number of small cubes with yellow paint on?
What interesting discoveries can you make?

If I gave you a list of decimals, you might find it easy to put them in order of size.
But what about ordering fractions?
A man called John Farey investigated sequences of fractions in order of size - they are called Farey Sequences.
The third Farey sequence is shown here.
The third Farey sequence lists in order in their simplest form, all the fractions between 0 and 1 which have denominators 1, 2 and 3.

The fourth Farey sequence is shown here.
The fourth Farey sequence lists in order in their simplest form, all the fractions between 0 and 1 which have denominators 1, 2, 3 and 4.

Can you find the fifth Farey sequence?

Take any four-digit number, move the first digit to the 'back of the queue' and move the rest along.
For example 5238 would become 2385.
Now add your two numbers.
Is the answer always a multiple of 11? Can you prove it?
What happens when you do this with three-digit numbers? Five-digit numbers? Six-digit numbers? 38-digit numbers ... ?
Prove your findings!

You could try things like:
2000+10 or 201 x 10 or 500+1000+500+5+5
or even

See what different questions you can find and compare yours with others. Perhaps you can make up one that is easy, one that is harder and one that is very hard.

Don't forget you might like to use calculators or computers!