Algebra is weird

Algebra is weird. This is a very short post in which I’ll show that algebra is a little odd.

Let’s start with a sum:

1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{2^4} + \dots

Where the dots mean that we keep on adding half the amount we just added forever. We take an algebraic approach to this sum, set

x = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{2^4} + \dots

So,

2x = 2 + 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{2^4} + \dots

2x= 2 + x

We solve this to see that x = 2. 

This gave rise to a terrible joke. An infinite number of mathematicians walk into a bar.  The first orders a pint of beer, the second a half pint, the third a quarter of a pint and so on.  The barman sighs and just pours out two pints telling them to share.

I don’t think x = 2 is a surprising result, its nice, but not surprising. Let’s try another:

1 + 2 + 4 + 8 + 2^4 + \dots where this time the dots mean we keep doubling the last term and adding it on.  Taking the same algebraic approach, we set

y = 1 + 2 + 4 + 8 + 2^4 + \dots and double each side,

2y = 2 + 4 + 8 + 2^4 + \dots

2y = y-1

We solve this and find that y = -1

Algebra is weird.

Engineering in Action

This post first appeared here.

Last Monday the inaugural Engineering in Action enrichment day was held at the Emmanuel Centre in London. Hundreds of year twelve students from across the country came to hear from and ask questions of five brilliant engineers.

First up was Anna Plozajski, a materials engineer working at the prestigious Institute of Making. It was humbling to hear about Anna’s incredible career from building a rocket car to working at NASA.

Second on the Stage was Hugh Hunt, a vibrations engineer at The University of Cambridge. Hugh’s highly entertaining talk had a huge number of demos, confirming ideas about forces and couples.

After lunch students heard from Amy Wright, a civil engineer at Farrans Construction who worked on the recently completed £117million Northern Spire bridge. Amy began her session with ten students on the stage deciding where to hide in a zombie apocalypse. In the second half of Amy’s presentation she talked about the Northern Spire bridge and you couldn’t help but be impressed by the sheer scale of the project.

Next up Paul Hellier from University College London took to the stage. Paul is a chemical engineer who, with the help of some students, performed two experiments investigating how efficiently bio-fuels made from crops vs. those made from algae burn.

Finally, Sam Rogers from Gravity Industries described the Iron Man style human flight suit he helped develop. It was fascinating to see the way Sam used 3D printing techniques to create jet packs. Needless to say, there were an awful lot of questions for Sam after his talk.

Steve Cross compered the day brilliantly, ensuring we had lively Q&A sessions and supporting all of the engineers.

Most Education in Action enrichment days build on the curriculum of an academic subject like Biology in Action or Maths in Action so it was exciting to have the opportunity to do something a bit different with this new, vocational, Engineering in Action.

We had longer Q&A sessions than usual after every speaker. Engineers like Hugh Hunt spoke about getting into university to study engineering and others like Amy Wright talked about work placement schemes at large engineering firms.

One teacher told me that two of her students want to be civil engineers after Amy’s talk because now they know what it involves.

We’ll be back next year with another day of Engineering in Action!

#MathsConf19

Last Saturday (24th June) La Salle Education held a national teacher’s conference #MathsConf19. It was a great event where I enjoyed attending talks and speaking to teachers about Maths in Action enrichment days.

I particularly enjoyed a talk on Making Mathematicians by Kate Milnes, a teacher of maths for over twelve years, at #MathsConf19. Kate argued that in most jobs we are required to solve problems, we start with a scenario, ask our self questions and then pursue a line of enquiry. She then shared a lot of ideas about how teachers can set up their maths lessons to develop these problem-solving skills in students.

Two ideas that Kate kept returning to were conjectures and generalisations, these excited me because of my background in pure maths. Kate used the 1089 trick to illustrate her ideas:

  1. Take any three-digit number where the first and last digits differ by 2 or more.
  2. Reverse the digits, and subtract the smaller from the larger one.
  3. Add to this result the number produced by reversing its digits

The resulting number, subject to a few conditions, is always 1089. To begin with a class of students has a go at the arithmetic with their favourite three digit numbers. They’re then encouraged to conjecture whether it works for all three digit numbers. Then they prove it.

Next we generalise, the class are asked to do steps 1,2 and 3 above with a four digit number and it turns out (again, subject to a few conditions) that the resulting number is always 10890. The pupils prove that. Then the students can conjecture what will happen if we use five digit numbers. It turns out that it’s not 108900. But that’s OK because not all conjectures turn out to be true.

We can generalise again, what happens if we follow steps 1,2, and 3 with a three digit number base seven? Usually you get 1056. The students can conjecture what happens in base eight, and then base n. They can have a go at proving it too!

This investigation is really fantastic. Never mind the students, I had a whale of a time conjecturing, calculating and proving during Kate’s session! You can generalise this task so many different ways, each time encouraging students to ask themselves questions, and pursue a line of enquiry.

I often hear the question “what’s the point” when it come to maths, pure maths in particular. Sometimes my answers feel a bit hollow but the next time I’m asked that question I will bring up 1089.

17 camels

Recently I had the pleasure of attending a conference dedicated to the History of Recreational Mathematics where David Singmaster gave a wonderful talk on the problem of the 17 camels (and a related problem on 13 camels).  The ideas below all came from his talk.

In a common version of the 17 camels problem a Sheikh dies, leaving his 17 camels to his 3 sons. The Sheikh decrees that his oldest son shall inherit one in two camels, his middle son shall inherit one in three camels and his youngest son shall inherit one in nine camels. The three don’t know what to do so they ask a wise man who advises them.  What does he advise?

The wise man loans the sons his camel, thus making 18 camels to divvy up. The oldest son takes 9 camels, the second son takes 6 camels, the youngest son takes 2 camels and the wise man rides off on his original camel.

I had fun thinking up a similar version of the puzzle this morning:

After the sad death of farmer John a lawyer visited his four children to divide up his estate.  John left 1/2 his estate to his firstborn, Alastair,  1/3 of his estate to his second child, Bertha, 1/10 of his estate to his third child, Colin, and a meagre 1/18 of his estate to the youngest child, Dorothy.

The lawyer announced that John was not the most successful farmer in the world (or even in West Dorset) and the only item to be divided up was a herd of 89 cattle. After a few moments of silent contemplation all four children leapt up and began arguing about how many cows they should each get and why their father had devised such a stupid way of dividing up his earthly belongings. 

Bertha was mid-flow, bellowing words that I will not repeat here at Alastair when a wise old cow from a neighbouring farm entered the unhappy scene. She stepped forward and uttered,

“Stop this unnecessary beef and let me be of assistance”

“How, how?” exclaimed the children

“youuuuuuu cud add me to your herd and divide your father’s estate again”

After wondering quite how wise this wise old cow actually was the children agreed to add the cow to their herd.  The lawyer smiled and bequeathed 45 cows to Alastair, 30 cows to Bertha, 9 cows to Colin and 5 cows to Dorothy, just as their father had requested. 

The children were astonished to see the wise old cow hadn’t been assigned to any of them. The lawyer laughed and walked off with the wise old cow and he later made millions touring the world with his amazing talking cow act.

The classic story about a father how distributes his 17 camels as 1/2, 1/3, 1/9 works because

Just for fun we might ask, suppose there are n heirs, how can we set a version of the camels? We need to find natural numbers

 such that

This gives you a handle on how to construct amusing examples like the farmer John example above.  The number of possible solutions grows rather quickly as you can see here.

Puzzle for Today

You could describe me as a ‘puzzle enthusiast’.  I look back fondly at the month I took out of my PhD to do a GCHQ Christmas Quiz (although I’m still annoyed that I didn’t win the darned paperweight). There is joy in discussing difficult problems with like-minded people, bonding whilst trying out different ideas and, hopefully, celebrating together when a solution is found.

Every morning Radio 4’s Today programme issues a Puzzle for Today. These usually take the form of a mathematical or logic puzzle and their difficulty varies substantially. An index of recent puzzles can be found here.  Over the last few weeks there has been a discussion online criticising these puzzles. One supposedly poor puzzle was Puzzle No. 464, set on Tuesday 23 April 2019. 

Puzzle No. 464

A drake, rook, duck and cuckoo are playing a personalised game of cards, but are arguing whether an ace is worth 1 or 11. The drake says his hand of cards are worth 8, the rook says hers are worth 4, the duck says her are worth 6, and the cuckoo says his are worth 3. What value should the ace be set at, to stop the birds being angry?

You can find the solution to Puzzle No. 464 further down this post.

Part of my job involves writing interesting puzzles for school students and I want to collect my thoughts on puzzles and puzzle-setting here. Let’s begin with a question I found here.

P: Chris came to a market to sell some eggs. Three people bought the eggs. The first buyer took half the eggs plus half an egg. The second buyer took half the remaining eggs plus half an egg. The third buyer bought what was left over: 1 egg. How many eggs were there initially?

You can find the answer to P at the bottom of this post. Suppose question P appeared in a maths text book, I would consider it just an algebra question. But, were P to appear in a newspaper, I would think of it as a puzzle.  Context matters.

When setting a puzzle, I start with the audience. What prior knowledge can we safely assume of our audience? A cryptic crossword, for example is ill suited to six-year olds because a six-year-old is very unlikely to understand the methods need to solve the puzzle. I ask myself how able is the audience and what will challenge them?

I think about how much time I want the audience to spend finding the answer. I ask students to complete four puzzles during a one-hour lunch break which gives me a very clear target.  A puzzle should always be both solvable and non-trivial. A lot of satisfaction comes from having solved a genuine problem and exerted some effort.

I always strive for clarity, both in the question and the answer. Puzzle-setters should be humble enough not to expect the audience to know what they are thinking. This is much more difficult skill than you might imagine!  Trying to spot ambiguities in a puzzle requires empathy.

Here is the answer to Puzzle No. 464:

1.

This is because the birds have assigned the values of their hands of cards to the sum of the letters in their names (after all, it is personalised), where each letter has its own numerical value. Fortunately we need not attempt to calculate the values of individual letters because, through cancelling the letters in their names, one can see that: Drake-duck-rook+cuckoo=ace. Using the given values, this means ace=8-6-4+3=1.

Puzzles that are just tricks, or jokes, or riddles often leave the reader feeling cheated or worse, stupid. This, I believe, is why Puzzle No. 464 frustrated some people. The fundamental problem with Puzzle No. 464 is that it isn’t suitable for the audience. It would be a great fit for a GCHQ quiz book where the riddles flow, but not for the majority of Radio 4 listeners.   

Radio 4 should ensure that every Puzzle for Today is clear, solvable and have a single solution.  They should also either steer away from riddles or pose a ‘Riddle for Today’.

The huge and varied audience of Today means that it is extremely hard for the problem setters to answer the big question: who is this puzzle for?  They cannot easily assume a level of prior knowledge or a suitable difficulty, so perhaps we should cut them some slack.

Answer to puzzle P:

7