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# Ordered Sums

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### Doodles

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Age 14 to 16

Challenge Level

- Problem
- Student Solutions

Let *a(n)* be the number of ways of expressing the
integer *n* as an ordered sum of 1's and 2's. For example,
*a*(4) = 5 because:

4 = | 2 + 2 |

2 + 1 + 1 | |

1 + 2 + 1 | |

1 + 1 + 2 | |

1 + 1 + 1 + 1. |

Let *b(n)* be the number of ways of expressing *n*
as an ordered sum of integers greater than 1.

(i) | Calculate a(n) and b(n) for
n 8. What do
you notice about these sequences? |

(ii) | Find a relation between a(p) and
b(q). |

(iii) | Prove your conjectures. |

Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?