### 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…]

### The Infinity of the Primes

Euclid’s Elements, Book IX, Proposition 20. Theorem : Prime numbers are more than any assigned multitude of prime numbers. That is, there are infinitely many primes.   Proof : Suppose that there are primes : [more…]

### Distinct and Odd Partitions

Euler’s Partition Theorem: The number of distinct partitions of an integer, , is equal to the number of odd partitions of that integer : , for all . Euler gave us a proof which is [more…]

### Computing Partitions

Euler gave us the following recursion formula for  via his Pentagonal Number Theorem : Note the generalised pentagonal numbers in the above. Define   if and let .     This is the formula that [more…]

### Shrove Tuesday

Pancake batter 4 oz plain flour A pinch of salt 2 large eggs 7 fl oz milk 3 fl oz water 2 tablespoons of melted butter A little extra butter for cooking the pancakes.   [more…]

### Unrestricted Partitions

Some may have noticed that I have been giving a representation of the countdown clock. The first line is the fundamental theorem of arithmetic.  That is, every integer can be represented by a unique factorisation [more…]

### Euler’s Pentagonal Number Theorem

Euler, : In considering the partitions of numbers, I examined, a long time ago, the expression in which the product is assumed to be infinite.  In order to see what kind of series will result, [more…]