More Mathematical Snippets
Tim Honeywill
(honeywillt@khviii.net)
MEI Conference 2015
This really was lots of snippets! Some I showed in my session, others were left for you to browse
through at your leisure.
Many of my powerpoints do not lend themselves well to being posted on a website, so the easiest
thing is to e-mail me at the above address and I will e-mail you my resources directly.
Here is the list of what I included (pieces in bold where shown in the session):
1. ‘My PIN is the last 4 digits of Pi’ – a powerpoint of funnies for people as they walked in,
admittedly some more funny than others!
2. End Game – a simple but effective game you could play with any secondary school year
group.
3. Counters on a rectangle – a simple puzzle with an ingenious proof that could be accessible
to even strong Year 7 students.
4. Magic Square – another simple effective ‘mathemagic trick’ that you could get any Year
group to create for themselves and to try out on families and friends.
5. Divisibility Tests – at school we all get used to describing the standard tests for divisibility by
2, 3, 4, 5, 6, 8, 9, 10 and 11 (with 7 at a push!), but we would be wrong to assume that there
are no simple tests for divisibility by other numbers. Although I choose very big numbers in
my powerpoint, with much smaller numbers you could tell Year 7 students about some of
these divisibility tests, just as a matter of interest and to make them aware of their
existence. To understand the explanation of why the divisibility tests work you probably
need to be an advanced A Level student.
6. Noughts and Crosses Version 2 – I don’t actually say what the rules are in the powerpoint!
Inside each noughts and crosses section is another noughts and crosses board! If you win a
noughts and crosses board then you win that section in the bigger noughts and crosses
board. There is also a great catch that where a player plays determines which section you
have to play in next; if a player plays in the middle left square of a noughts and crosses
board then the other player has to play in the noughts and crosses board that is in the
middle left section of the bigger noughts and crosses board (if that board is already full then
he/she can choose where to play). If someone is lucky to get a row (horizontal, vertical or
diagonal) of won smaller noughts and crosses boards then he/she wins; otherwise you play
until every space is filled and count up who has won the most noughts and crosses boards.
This took off like a storm with sixth formers at my school. Is this just too complicated to
analyse mathematically or does anyone fancy exploring this with me?!
7. Hungarian Race Driver – I have shown this to Year 7 students and to sixth formers. There are
lots of animations and so you probably need to e-mail me if you want to have a copy of this
talk to show to students. It is a good example of a problem that is very hard to solve unless
you allow the possibility of the completely ridiculous!
8. Holy Sphere – A surprising puzzle result which you could put to Year 13 students. Thanks to
Colin Wright, I now know of an approach to the same problem that you could put to
advanced GCSE students; please e-mail me if you want to know more.
9. Infinite Checkerboard – from John Conway, this puzzle fascinated me. I have enjoyed talking
about this to students of all ages, although understanding what Conway did would be a good
investigation for strong A Level students (end of Year 12 or above). I only sketch the proof at
this stage, since Bernard Murphy talked about this result at a previous MEI Conference, but I
do mention extensions to this problem and ask if anyone wants to explore with me!
10. Math Tricks – this was an App I got on my phone, designed to improve your mental
arithmetic, teaching you hundreds of rules and then getting you to answer endless questions
that use each rule. This was fun for me but it frustrates me that so often people are just told
what to do, with no explanation at all of why it works. Here are some of the tricks on the
App that I have selected, for students to understand WHY the tricks work. I hope to give this
out when teaching Year 12 Proof.
11. Set Bingo – this is an idea I have for playing a card game with students who have just been
taught the various different pieces of set notation. This is still work in progress but the
students seem keen. Please e-mail me if you want to get involved.
12. Mechanics and the Compound Angle Formulae – I like it went different pieces of
Mathematics agree with each other. Here are two ways in which you can quickly see the
compound angle formulae appearing.
13. PP…P Primes – a number is PP…P prime if it’s first digit is prime, its first two digits form a
prime number, its first three digits form a prime number, etc. …, and it is itself a prime
number. How many PP…P primes do you think there might be? Finitely many or infinitely
many? Find out in this powerpoint!
14. Happy Numbers – to any positive whole number you can add together the sum of the
squares of the digits… and then you can do the same to the answer you get… and then the
same again… and so on…
What happens? Welcome to the world of Happy and Unhappy Numbers!
15. Summing Reciprocals – another remarkable result that would stretch the proof skills of A
Level Further Mathematics students.
16. Digits In A Circle – a fun result that is less demanding but would be good for Year 12
students to prove.
17. 100 Prisoners Problem – this is a powerpoint of one of my favourite problems, designed for
showing to students. It has hundreds of animations so, again, it probably would be best to email me if you want a copy. One day, I hope to add a bit that proves a bit more of what
happens. I would show this problem to any secondary school Year group; how much I discuss
proof that the strategy works depends on what Year group I am with!
18. Peppa Pig – following watching an episode of ‘Bouncy Ball’ with my neice, I thought that,
from the trajectory of the ball, there is no way that the gravitational constant is that of
Earth. I would like to set an open-ended (and light-hearted) investigation for students at the
end of Year 12 where I show them the video of ‘Bouncy Ball’ (probably not all of it!) and
simply ask them “On which planet does Peppa Pig live?”. The powerpoint shows my efforts,
though very different approaches could be used by other people.