Take an A4 piece of paper.
Fold it a few times.
Then punch a hole through the paper.
Before you unfold your paper, can you draw on a fresh piece of paper where you think the holes will be?
Then can you fold a piece of paper and punch it to make the holes look like this:
Then can you fold and cut a piece of paper to make it wider then it was originally? Can you cut in such a way that you can step through it?
Similar ideas can be found here. Solution to stepping through it here.
On dotted paper we explored circles of different radius, counting the dots on or inside the circle. Brilliantly the number of dots is very close to the area of the circle.
We wrote this program in Python to check the ratio of dots to area for different size circles.
Challenge 1
Imagine you have an endless supply of 5p and 7p coins. You could make exactly 20p with four 5p coins. Or you could make exactly 19p with two 7p coins and a 5p coin.
What is the biggest amount that you cannot make? Can you explain why you can make all amounts after this one?
Challenge 2
The biggest number you cannot make given coins of value x and y is called the Frobenius number.
Can you think of a pair of numbers that wouldn’t have a Frobenius number?
What must be true about the two numbers for them to have a Frobenius number?
Challenge 3
Can you come up with a formula for the Frobenius number, if it exists, for two numbers x and y?
Solutions here.