### Two Squares and Four Squares

Diophantus of Alexandria wrote a collection of books called Arithmetica in the 3rd century AD.  He noted that natural numbers of the form cannot be expressed as the sum of two square numbers.  He also [more…]

### Heads, Tails And Beans |S|=6

This is an unintentional detour for our little course. I didn’t plan to investigate the factorial function in so much detail, but it came about quite by accident (or stupidity) as we’ll see in later [more…]

### Heads, Tails And Beans |S|=5

Here we’ll look at some more probability calculations before moving on to permutations, combinations and the binomial theorem. More Probability… If you recall from the last part, we used set notation to describe a general [more…]

### Heads, Tails And Beans |S|=4

So far, we’ve looked at sets and how they contain information that we wish to count in some way. How the power set $$\wp(S)$$ can count all of the possible sub-sets of a given set [more…]

### Heads, Tails And Beans |S|=3

Subsets Let’s suppose that set $$S$$ contains a full deck of playing cards. Therefore, $$|S|=52$$. We can split the cards into four suits: $$♠,♣,♥,♦$$. Let’s make a set $$T$$ that contains only the clubs suit [more…]

### Heads, Tails And Beans |S|=2

Before we can start to solve some counting problems we need to know a little of combinatorics’ notation. Luckily, a few simple rules should get us started. Don’t worry if it doesn’t make sense, that [more…]

### Heads, Tails And Beans |S|=1

Mathematics is not everyone’s cup of tea – fair enough. The Monday morning ‘double maffs’ articles by our resident mathematician, OldTrout, can present some GP’ers with a feeling of bewilderment. Myself included. I’m no mathematician. [more…]

### Proof by Necklaces

A proof of Fermat’s Little Theorem using necklaces. , where is prime and is any integer with gcd.   Multiply both sides of the congruence by : Which gives : Subtract from both sides : [more…]

### The Confusing Bit

When you do mathematics you spend much of your time being confused. We can go further with our counting and count the number of necklaces and bracelets with a particular colouring. When we counted the [more…]