F(n) and M(n) are defined in a recursive, or doubling-back-on-themselves, way as:

F(n) = n─M(F(n─1))

M(n) = n─F(M(n─1))

F(0) = 1; M(0) = 0.

• By using a spreadsheet program with VLOOKUP, or by hand, calculate values of F and M up to n=55.

For most n, F(n)=M(n).

• What can you find about when F(n)≠M(n)?

A Freddo and fame for every good attempt at even a partial solution got to Mr Thomas by late June 2018.

**Maths prize 2 March 2018: recursive definition**

The function g(n) is defined in a doubling-back-on-itself way, or recursively, by:

g(0)=0 and g(n) = n − g(g(n−1)) for n>0.

• Prove by induction that 0<g(n)<n for all n>0.

• Calculate g(n) for n=1 to 21 (VLOOKUP in Excel may help you with that).

The tree above is values from a similar recursive definition, set out by writing below each number m the values n for which g(n)=m.

• How would the tree for the values you have calculated compare?

• What famous sequence do you see in the tree?

• What recursive definition would produce the tree on the left?

Howard Tran won the prize with the best effort at a solution: see https://mathsmartinthomas.wordpress.com/2018/02/15/maths-prize-2-march-2018-recursive-definition/.

**Maths prize 9 Feb 2018: is MU a word?**

A language has an alphabet of 3 letters, M, I, and U. Words are formed in this language by the following rules (where “x” and “y” represents any string of letters, not necessarily words).

1: MI is a word

2: if xI is a word, then so is xIU

3: if Mx is a word, then so is Mxx

4: if xIIIy is a word, then so is xUy

5: if xUUy is a word, then so is xy

Question: is MU a word?

(This puzzle is taken from Douglas Hofstadter’s book Gödel, Escher, Bach, where it is used introduce thinking about some of the maths important for computer science).

The prize was won by Mohaned al-Bassam.

Maths prize 24 Jan: how tough is the smartphone?

Maths prize 24 Jan: how tough is the smartphone?

A new smartphone will survive being dropped several storeys. But how many? A 64-storey drop will smash it. Given a 64-storey building, how can you find the exact maximum drop in just six experiments? And how many phones do you need for that?

A Freddo and fame for every solution, or good attempt, to Mr Thomas by Friday 24 January 2018.

Archive of previous prize puzzles: https://mathsmartinthomas.wordpress.com/2017/12/20/maths-prize-puzzle-archive-2014-2017/