### Factoring a Million

In how many ways can the number 1 000 000 be expressed as the product of three positive integers?

### Thank Your Lucky Stars

A counter is placed in the bottom right hand corner of a grid. You toss a coin and move the star according to the following rules: ... What is the probability that you end up in the top left-hand corner of the grid?

### Card Game (a Simple Version of Clock Patience)

Four cards are shuffled and placed into two piles of two. Starting with the first pile of cards - turn a card over... You win if all your cards end up in the trays before you run out of cards in one pile. What is the probability that you win?

# Sheffuls

##### Stage: 4 Challenge Level:

Can you write down all the ways that you can reorder the letters in sheffuls to make the word shuffles ?

To answer questions like this you need to have a good way of representing reorderings. Our aim is to introduce you to a useful way of doing this called cycle notation that will help you manipulate these permutations on paper. A permutation takes an ordered set of elements and shuffles them into a new order.

Work through the examples below, and then you should be able to solve our problem.

You'll find the shuffle interactivity helpful. There is also a short tutorial that explains how to create and manipulate shuffles.

The picture shows a simple shuffle which can be written as (1 4 3 2) in cycle notation. Think 1 goes to 4, 4 goes to 3, 3 goes to 2, and finally 2 goes back to 1 forming a cycle.

Not every shuffle can be represented with just one cycle. Here are two that require 3 cycles.

However, shuffles can always be represented by a list of cycles. The diagrams might help you work out why this is so.

There is one special case - the identity shuffle which does nothing. Call this (1).

The permutation (2 3) reorders the second and third elements, leaving everything else unchanged. In the shuffles interactivity this same permutation can appear as different length shuffles like this:

Try using the interactivity to make these permutations:

(1 2)(3 4)
(1 4)(2 3)
(1 4 2)
Now try making these. There's more than one way to write a cycle, but the interactivity always starts with the smallest number.
(3 2 1)
(4 2 3 1)
A little more practice! How many ways are there of writing this shuffle (1 2 3 4)(5 6 7)? What could (1 2)(2 3) mean? How about (2 3)(1 2)? Can you simplify these expressions? Can you create them in the interactivity?

Now, back to the main question:

Can you write down all the ways that you can reorder the letters in sheffuls to make the word shuffles using the cycle notation?