# Can you crack the Puzzle for Today?

Academics from the Faculty of Actuarial Science and Insurance at Cass Business School have been taking part in Puzzle for Today on the Today programme on BBC Radio 4.

The Puzzle for Today slot, which is on air around 6.45 am from Monday to Saturday, involves a mathematician, statistician or actuarial scientist challenging listeners to solve a mathematical puzzle, often based around a current event in the news cycle.

Dr Russell Gerrard, David Hargreaves and Robin Michaelson have all recently set puzzles – see if you can solve them below.

## Dr Russell Gerrard: The Magentas

Puzzle

Dissatisfied politicians in both Blue and Red parties have decided to set up the Magenta party.  When a former member of the Blue party joins the Magentas, there is a 90 per cent chance that the next person to join will also be a former Blue.  When a former member of the Red party joins the Magentas, however, the chance is 75 per cent that the next new member will also be a former Red.  As the Magenta party grows, what is the proportion of members who used to belong to the Blue party?

This is really a question about the equilibrium distribution of a Markov chain.  To find the answer you say: Think of a particular new member of the Magenta party, such as the first one after 1st March.  Let b be the probability that they are an ex-Blue, r for a former Red.  Then the probability that the second new member after March 1st comes from the Blue party is b * 9/10 + r * 1/4: but this must also be equal to b.  Therefore r = 0.4 b.  Since they sum to one, we must have b = 5/7, r = 2/7. So, in the long run, five in seven Magenta party members used to be Blues.

## David Hargreaves: Fairground game

Puzzle

A fairground game involves rolling a die as many times as you want. You win if the total of all your rolls is a number you have chosen before you play. What number should you choose?

Six.  It is easier to get to two than one, as there are two routes:

• Two direct (one in six chance) and
• One then one (one in 36 chance)

By extension, it is easier to get to three than two, four than three, five than four and six than five.

However, there is no direct to numbers seven and greater, but it is certain that there will be a first time you are within six of any such larger number so the chance of hitting a number greater than six is a weighted average of the chances of hitting each of the numbers one to six and so cannot be more likely than the most likely of the numbers one to six.  Therefore, the best number to pick is six.

## Robin Michaelson: The professor's daughters

Puzzle

A maths professor had invited a colleague to his home, and as the two of them were walking there, the colleague said to the professor: “Do you have any children?”

“Yes,” said the professor. “I have three daughters.”

The colleague said: “what are their ages?”

The professor said: “the product of their ages is 36.”

The colleague: “I cannot tell their ages from that.”

The professor said: “the sum of their ages is one more than the house we are passing.”

Again the colleague said: “I cannot tell their ages from that.”

The professor said: “the eldest has just started piano lessons.”

With that information, the colleague immediately knew their ages.

What are the daughters’ ages?

The answer is nine - two - two and it is clever because if you break down all the possible combinations, ie:

• 36 - one - one (improbable!)
• 18 - two - one (at best unlikely)
• 12 - three - one
• Nine - two - two
• Six - six - one
• Six - three - two
• Nine - four - one
• Four - three - three

Two sets (nine/two/two and six/six/one) have identical sums, which is why the colleague could not deduce their ages from the passing house until he learnt that "the eldest" was taking piano lessons, whereas six/six/one does not have an eldest, hence the answer is nine - two - two.

Dr Russell Gerrard, Associate Dean for Learning and Teaching

Russell Gerrard was a Wrangler at the University of Cambridge and remained at the same institution to complete a PhD in Stochastic Processes.  After a year at the University of Sussex he spent nine months at Moscow State University, followed by six months at the University of Zurich.  He subsequently joined City, University of London in the Department of Mathematics and currently works in the Faculty of Actuarial Science and Insurance of Cass Business School, which forms part of City. His research focuses on applications of probability and statistics, primarily to problems of an actuarial nature.  He has performed as a Principal Examiner for the Institute and Faculty of Actuaries and has served as the Associate Dean for the undergraduate programme at Cass.

Robin Michaelson, Visiting Lecturer

Robin Michaelson graduated from Corpus Christi College, Oxford, and is a Fellow of the Institute of Actuaries, having spent his working career in life insurance and reinsurance, in both the UK and Australia.  In retirement, he is a Visiting Lecturer on Actuarial Practicality at the Cass Business School, in the Faculty of Actuarial Science and Insurance.  He has always been interested in mathematical puzzles and teasers, and in how they can relate to everyday life."

David Hargreaves, Visiting Lecturer

David is a mathematician who graduated from Cambridge University in 1991 before embarking on careers in the Police, teaching and the actuarial profession. He returned to teaching in 2013 becoming a visiting lecturer at Cass Business School where he has taught across a wide range of actuarial subjects.  Outside of work, David has developed a computer program to play poker (which is the only piece of mathematics, he has done, to interest non-mathematicians) and is trying to eliminate the housing crisis by getting politicians to understand the interaction of housing benefit and fractional reserve banking (currently with no success).

## Find out more

Find out more about Actuarial Science and Insurance at Cass Business School here.

Join the conversation #CassExperts