As there will not be an NRICH magazine in August we have included a few extra challenges set by some young NRICH members. You can send in your solutions in the usual way. Thank you very much Madras College for these challenges.

Here are two challenges based on the number 1999, both from Madras College which is a comprehensive secondary school founded in 1833 by Rev. Dr. Andrew Bell and located in Fife, Scotland. From their website , I learned that Dr Bell was a mathematician. He would be very proud indeed of the mathematical achievements of this school today. He served as Chaplain with the East India Company in Madras and very successfully used older students to help to teach the younger ones.

The first challenge is to use the digits 1, 9, 9 and 9 together with mathematical operations to form as many whole numbers as you can. You will see solutions up to the total 140 on the Madras College Maths Department website (and there is lots more of interest there too). If you have new solutions to this challenge then please send them to Madras College rather than to NRICH. The same sort of challenge will be very hard next year!

Secondly, class 2YP at Madras College carried out their own investigation based on the problem called SCORE in the NRICH June Six Problems. They came up with some intriguing and surprising results. One problem they set themselves was to find the number of ways of writing 1999 as the sum of three odd numbers and we leave it as a challenge for you to find out how many ways this can be done. Do send your solutions to NRICH if you wish. The account of 2YP's work will be published on 1st September.

A Year 9 pupil from Dr Challoner's Grammar School, Amersham (United Kingdom) sent two challenges. Frogtastic may not be entirely new to you but Theodor's Birthday will be sure to set you thinking.

The Editor

Rules:

- You have to swap the x and y frogs over to the other side.
- A frog can only jump over a frog of a different kind.
- A frog can only move to one square at a time.

x | y |

x | x | y | y |

and so on ...

Number of frogs | Number of moves |

1 | ? |

2 | ? |

3 | ? |

4 | ? |

5 | ? |

TASK:

- Complete table above.
- Did you find any relationship between the number of frogs and the number of moves?
- Consider the same problem with different numbers of x and y.
- Have equal number of x and y but vary the gaps between.
- Consider different numbers of x and y and vary the gaps between.

Could you find any other relationships in this game?

On 19th of May it was Theodor's Birthday. On that day he had a brilliant party, however his parents, uncles and auntie's couldn't make it to the party, so on the day after they had dinner together.

- Theodor
- Thiruthdevy (his mother)
- Vallipuranathan (his father)
- Loga (his auntie)
- Ganesh (his uncle)
- Bavananthan (his uncle)

All of the six people sat around a circular table

TASK:

- If Theodor sits in seat T , how many ways can the others sit?
- If there were only 2 chairs and 2 people sitting how many ways can they sit?

3 chairs with 3 people sitting...

4 chair with 4 people sitting...

and so on... - What is the relationship?
- What if there are
*n*chairs with*n*people? - If any of the 6 people at the dinner do not wish to sit next to each other, how many ways of sitting are there?

3 people refusing to sit next to each other...

4 people refusing to sit next to each other...

and so on...