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…]
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…]
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…]
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…]
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…]
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…]
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…]
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, : 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…]
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