Answers to last week's questions.

Last week I posted some lateral thinking and trick questions. The idea being that, when I do outreach workshops to younger audiences, they will be less worried about being wrong if they realise that everyone will be as wrong as they are. Again, I would like to reiterate the point that this is nothing about making the students feel stupid, but rather empowered as they should not be worried about make "silly" suggestions.

________________________________________________________________

________________________________________

1. The questions starts off with "You're driving a bus". Thus, the eye colour of the driver is whatever yours is.
   
   The idea behind this question is to emphasise the point that to produce an answer you first have to understand what information you need from the question.


2. If you take 307 bananas from 429 bananas, how many bananas do you have? You have 307.


3. All months have 29 days.

Questions 2 and 3 deal with the idea of clarity of communication. When answering a question in mathematics you have to clearly define your terms and your assumptions.


4. The probability that exactly five are in the right envelopes is zero. If five are in the right envelopes the sixth must be two.
   
   Here, again, we are dealing with the idea of taking the important information from the question. The most important word in the question is "exactly". Thus, any solution we generate should be weighed up against the requirement.


5. Each dog takes five days to dig a hole. So ten dogs will take five days to dig ten holes.


6. The full stop at the end of the sentence is the smallest circle.

By now the students should realise that the questions are trick question, so questions 5 and 6 teaches them not to be too hasty with their answer and to think carefully even when the answer appears obvious.


7. Tuesday, Thursday, today and tomorrow.

This question asks them to find a solution to the seemingly impossible. The idea being that mathematicians need tenacity when working with a problem. Many times you it will seem like the question is intractable but eventually you will find the right path.


8. Noah built the Ark not Moses.


9. All the numbers are divisible by two. The question does not ask for integer answers.

Questions 8 and 9, again, show the importance of reading questions carefully and fully understanding what is being asked. Trust me, as a mathematician, the hard part is not generating a solution, but rather, understanding what the question is asking.


10. And the final puzzle. Did you spot it? The first instruction you are given is to write your name in the square. Next to this instruction is a rectangle. The square is at the bottom :).

Tricky.

Below are a few of my favourite lateral thinking and trick questions. Whenever I do outreach work shops on higher mathematics for secondary school I often open with this as a 5 minute ice breaker. Hopefully, the kids will not do very well. The reason I say hopefully is because I want to emphasise that they do not know everything. This is not meant in a bad way, but an encouraging way. Over the course of the workshop I touch on subjects such as logic and topology which they will have never seen. By getting the audience comfortable with being wrong and not knowing the answer I hope to encourage them to ask questions and provide solutions, even if they turn out to be wrong. 

Fear of being wrong or asking "silly questions" is a common barrier to over come in a class room situation. The participants are surrounded by their peers and the last thing they want it to do is to seem stupid. However, the workshops flow better if everyone is willing to suggest their insights on which to build. 

So, without further ado, try the questions yourself. Only give yourself 5 minutes to try out all the questions and, of course, don't cheat. I'll post the answers next Monday, so you can see how well you did. Remember

Research is what you do when you don't know what you are doing

________________________________________________________________

________________________________________

Write your name in the square:

1. You're driving a bus that is leaving on a trip from A and ending in B. To start off with, there were 32 passengers on the bus. At the next bus stop, 11 people get off and 9 people get on. At the next bus stop, 2 people get off and 2 people get on. At the next bus stop, 12 people get on and 16 people get off. At the next bus stop, 5 people get on and 3 people get off.  Question:  What colour are the bus driver's eyes?


2. If you take 307 bananas from 429 bananas, how many bananas do you have?


3. In a leap year how many months have 29 days?


4. A secretary prints out six different letters and is in a rush so she randomly stuffs the letters into six envelopes going to six different addresses. What is the probability that exactly 5 letters are in the right envelopes.


5. If five dogs dig five holes in five days, how long will it take ten dogs to dig ten holes? 


6. Highlight, in some way, the smallest circle.


7. Name four days which start with the letter T.


8. If animals enter the Ark in pairs at rate of 30 pairs per day, how many days would it take Moses to get 360 individual animals on board?


9. How many numbers between zero and ten, inclusive, can be divided by two?

Score:

Mathematical poetry 2.

The following poem is about a certain Professor Felix Fiddlesticks:

F set the coins out in a row
And chalked on each a letter, so,
To form the words "F AM NOT LICKED"
(An idea in his brain had clicked).
And now his mother he'll enjoin:
MA DO LIKE
ME TO FIND
FAKE COIN
-- Cedric A.B. Smith

Now this may appear to not make much sense, but it is in fact a really clever solutions to the 12 coin problem:

Imagine you are given 12 coins, and a set of weighing scales. 11 of the coins have the same weight, but one has a different weight. To make matters worse, you do not know if it is heavier of lighter than the others. The problem is to find out which coin is different, and whether it is lighter or heavier, using at most three weighings on a pair of scales.

Before you see the answer and I explain the connection between the puzzle and the poem, have a go at solving the problem yourself. Here is a flash applet which allows you to play that game yourself:

http://www.mapsofconsciousness.com/12coins/

________________________________________________________________

________________________________________

Did you solve it? Did the poem help? If you managed to do it, give yourself a pat on the back. Otherwise, let me explain. Firstly, do as the poem says; on the coins write one of the single letters F, A, M, N, O, T, L, I, C, K, E and D. Now, we weigh the coins

MADO against LIKE

METO against FIND

FAKE against COIN

 

in each of the weighings we have two possibilities. Either, the pans balance or they don't. If any of the weighings do balance, we immediately know that all 8 coins are genuine and the dodgy coin is in the four we haven't weighed. Conversely, if the pans do not balance then we know that the four coins not on the scales are genuine.

As a specific example, suppose that, in our weighings, the right pan is always lower. We can then logically deduce that the coin "I" is the dodgy one and that it is heavier, since coin "I" is the only coin that appears in the right balance in all three weighings. If you are interested in the entire solution to this problem look at the last post in this Dr. Math forum post.

Maths laugh 4. Tasty, tasty pie.

An oldy but a goody.

Fractal fun.

As a Mathemagician (I didn't pick the name) I have been lucky enough to work with Prof. Marcus du Sautoy on a number of occasions. One of my favourite projects that I have worked on was when he asked me to do some illustrations for his most recent book, The Number Mysteries. Specifically, in Chapter 2, The Story of Illusive Shape, he wanted some pictures of fractals. So for those of you who have read the book you'll see some familiar figures below and for those of you who haven't then hopefully these will pique your interest and you will go and find out more.

Figure 1. Illustrations of how to measure the dimension of a fractal by covering it in boxes of different lengths.

Figure 2. The Koch snowflake is a very famous fractal. Normally, it is constructed using an equilateral triangle. Here, I alter the angle to make it isosceles. What do you think that might do to the dimension?

 Figure 3. The Koch snowflake is constructed by replacing the middle third of a triangle with a triangle a third of the size. Normally, all the triangles are taken to point outwards. However, above I randomise this process.

Figure 4. If you put three of the randomised Koch snowflakes together you get something that looks like a medieval map of Britain

If in the future I run out of things to say I may put up the codes so that you can also learn how to create such structures.

Mathematical poetry 1.

Now, I'm sure I could spend a long time gushing over the various ways the people have tried to encapsulate the beauty of mathematical ideas through inspiring prose. Indeed, there are such things called "Piems", which are used to remind the reciter as to the digits of Pi. Each word is the length of the corresponding number in the decimal expansion. Here is a very impressive example called "Poe, E: Near a Raven" (3.1415). Not only does it allow you to calculate Pi to its 740th decimal place but it has used Edgar Allan Poe's poem "The Raven" as the subject matter.

I could mention Fibs which, like a haiku, is formed by each line having an exact number of syllables. In this case the number of syllables correspond to the Fibonacci sequence. This form of poetry even has  its own journal.

I could even mention that more and more artists, muscians and writers are turning to mathematics in order find inspiration to create aesthetic pieces.

However, I'm not going to do any of that as this is the blog of the laughing mathematician. I'm here to take a lighter look at the mathematical world and that is exactly what I'm going to do. Here you will find mathematical ditties, limericks and even a poem that solves a logic puzzle! What more could you want?

________________________________________________________________

________________________________________

So without further ado, here are a couple of my favourite limericks.

The integral z-squared dz,

from 1 to the cube root of 3,
times the cosine,

of three pi over nine

equals log of the cube root of e. 
-- Attributed to Betsy Devine and Joel E. Cohen

A mathematician confided,
that a Moebius strip is one-sided.
You'll get quite a laugh,
if you cut it in half.
For it stays in one piece when divided.

-- Anon

Zombies.

One of my favourite pieces of mathematics that I have ever done has been about zombies. Now, as a mathematical biologist, I often wonder whether zombies come under my remit. Are the biological? They aren't really alive. But anyway, sit back and let me recount the story.

In 2009 P. Munz, I. Hudea, J. Imad and R.J. Smith? wrote a paper called "When zombies attack!: Mathematical modeeling of an outbreak of zombie infection". His conclusions were bleak as it seems that if the dead should rise, there will be no room for the living. As you might expect this was a huge hit with the science/media interface. 

Fast forward to the end of 2010 and Robert Smith? (yes, the question mark is actually part of his name) decided that the original paper was such a success that he'd like to generate a whole book about zombie mathematics. He sent out an email call to his mathematical colleagues, asking if any of them were interested in writing a chapter for a layman's biological mathematics book, using zombies as the hook to interest the general reader. Well, what could I do? In my life I care about three things; mathematics, teaching people about mathematics and zombies - in that order. So I jumped at the chance.

The book is scheduled to be completed April 2012, but here are some of the pictures contained in my article. I created the man with the gun but I have to thank Martin Berube for the zombies.