Thursday, 8 January 2009

A new symmetry object

This is to report that Ameli Gottstein, who won the Christmas Maths Quiz, has chosen as her prize to have a new mathematical structure, a group of symmetries in multi-dimensional space, named after her. (Others similarly honoured include the footballers Jermain Defoe and David Bentley.)

A description of Ameli's group will shortly appear. If you want to name one of these amazing mathematical structures after someone, all you have to do is make a donation to Marcus du Sautoy's favourite charity and Marcus will name a group for you. This makes an excellent present (I think) and is in a very good cause.

Tuesday, 6 January 2009

Christmas maths quiz - solution

Here are the answers to the Christmas Maths Quiz posted last month. The best entry received from a Greenwich undergraduate came from Ameli Gottstein, who wins the prize. Well done Ameli!

Many of the questions (2, 3, 4, 5 and 6) were taken from Peter Winkler's Mathematical Puzzles: A Connoisseur's Collection (A.K. Peters).

The answers were:

Q1. Identify the mathematicians whose names have been shuffled so that the letters are in alphabetical order.
a) AEHLST - Thales
b) AAHIPTY - Hypatia
c) AAHKMYY - Khayyam
d) DEE - Dee
e) AEFMRT - Fermat
f) AHILLOPT - L'Hopital
g) EELRU - Euler
h) ADEGMNOR - De Morgan
i) AAAJMNNRU - Ramanujan
j) ACDIR - Dirac
As a tie-break, can you suggest a candidate for the mathematician with the longest name in which the letters are in alphabetical order, so that (like one of the above) they would appear unchanged if their name were included in the above list?
Peter Rowlett wrote a computer programme to address this and provided a list of all mathematicians in the St Andrews website whose names have this property. The longest such name waa that of Edwin Abbott (author of the classic mathematical novel Flatland). Peter also found mathematicians whose names are in reverse alphabetical order - the longest this time was Rolle.

Q2. Can you find English words containing the following letters consecutively? a) WKW b) HIPE c) ZV d) HQ e) NSW
The answers I was expecting were AWKWARD, ARCHIPELAGO, RENDEZVOUS, EARTHQUAKE and ANSWER. Alternatives proposed included (c) JAZZVOCALIST and (d) MATHQUIZ, which I liked.

Q3. Which of the United States is closest to Africa?
Maine. (If you don't believe this, have a look at a globe! Map projections distort distances!)

Q4. There are several different time zones in the United States. New York on the east coast is normally three hours ahead of Los Angeles on the west coast. A phone call is made from an East Coast state to a West Coast state and it is the same time at both ends. How can this be?
One for Americans, I think. Florida is an east coast state and part of Florida (for example Pensacola) is in the Central Time Zone, while parts of Oregon (including Ontario) observe Mountain Time, so Ontario, Oregon and Pensacola are only an hour apart. The trick now is to call from Pensacola to Ontario between 2 and 3am on the morning in October when Daylight Saving Time ends, so that the clock has gone back an hour in Pensacola but has not yet done so in Ontario, and the time in each is the same.

Q5. There are 100 lightbulbs and switches, numbered 1 – 100, and 100 students. Initially each bulb is off. The first student switches on every light. The second then switches off every second light. The third student now changes the state of every third bulb, so 3, 9, 15,… are switched off and 6, 12, 18, … are switched back on. The fourth student changes the state of every fourth bulb, and so on. After the 100th student has done this, which bulbs are left on?
The ones which are left are exactly the perfect squares: 1, 4, 9, 16, ...

Q6. On average, how many times would you have to roll a fair die before all six numbers have appeared?
If we have already thrown n of the six numbers, then the expected number of rolls before a new number is thrown is 6/(6-n). So the answer to the question is 6/6 + 6/5 + 6/4 + 6/3 + 6/2 + 6/1 which is about 14.7.

Q7. Sasha and Siobhan are in the same class. They look alike, have the same birthday, were born in the same year, and have the same parents. Yet they are not twins. Can you explain this?
They are triplets.

Q8. Let n be a natural number. Show that there is an integer (non-zero) multiple of n which (in base 10) contains only 1s and 0s.
We use the famous Pigeon-Hole Principle (PHP). Consider the integers 1, 11, 111, 1111, 11111 ... up to the integer with (n+1) ones. There are n+1 such integers and only n possible remainders when you divide by n, so two of these integers have the same remainder. If they were 11111111 and 1111, for example, then the difference would be 11110000 which is a multiple of n of the required form.

Q9. Can you decipher this message? SZKKBSLORWZBGLBLFZOO
HAPPY HOLODAY TO YOU ALL - this used the Atbash cipher.