Tag Archives: puzzles

#NoticeWonder and Rational Tangles

Yesterday we held the first of this year’s Maths Experience days. We invite students in Years 10 and 11 from different schools onto campus for an intensive one-day program. Students find out about mathematical research, talk to professionals who use mathematics in their careers in some way, and participate in hands-on mathematics workshops. Importantly, they also meet and connect with other students who enjoy mathematics.

One of the activities I chose for this year was Conway’s Rational Tangles. I’ve previously written detailed notes about running the activity with pre- and in-service teachers.  For the Maths Experience, apart from the inherent fun of ‘playing’ with ropes, I wanted students to have a collaborative and authentic problem-solving experience. I introduced the activity as one that mirrors mathematical research — full of questions, puzzling moments, uncertainty, frustration and hopefully also joy. I emphasised that we might not solve the problem, but that the experience of working mathematically was our goal, which includes making wild conjectures and having out-of-the-box ideas!

In this post I want to highlight one addition I made to the activity described in my earlier blog post — the inclusion of the ‘Notice and Wonder’ prompt1.

I started the session by showing students the short video below, edited from one I found on Youtube by Tom Hildebrand. Specifically, I turned off the sound, cut out the whiteboard, and sped it up significantly. Then I asked the two magic questions: ‘What do you notice? What do you wonder?’ Take 70 seconds to watch the video, and see what you think.

Here is what they noticed.

Group A

  • They are trying to untangle the ropes.
  • One person hangs on to one end of the rope for the whole time.
  • They rotate 90 degrees clockwise.
  • There is a plastic bag.
  • Twist involves exactly two people and occurs in exactly the same position.
  • They untangle using exactly the same types of moves they used to tangle.

Group B

  • Four people holding two ropes.
  • Same person holding the same end for the whole activity.
  • When rotating, one person moves clockwise. (Later refined to each person moves one position clockwise.)
  • The twist movement always involves the two people on the right. The same position goes under each time.
  • There was some pattern they kept repeating.
  • They did some moves to get a knot. Then they did some more moves and there was no knot.
  • There was a bag.
  • There were four rotations before the bag appeared and eight rotations after.
  • Sometimes there is a different number of twists after a rotate.
  • A twist after a rotate goes ‘perpendicular’. (Not sure what that means!)

And here is what they wondered.

Group A

  • What’s the deal with the plastic bag?
  • What’s the deal with the teacher?
  • How did they decide when to stop tangling and start untangling?
  • How tangled was the rope?
  • What did the teacher and the student pass to each other? (Scissors.)
  • How did they work out how to untangle? (I explicitly prompted this question — although I’m sure they were all thinking it.)

Group B

  • How did they know how to untangle the ropes? Was it from memory?
  • What is the point of rotate? It doesn’t seem to change the rope.
  • Does the bag have something to do with the tangling?
  • Is it a proper knot? Or just a tangle?
  • What is the teacher doing?

There was more conversation that I didn’t manage to capture. (Next time I’ll record it!) Group A spent around 10 minutes on Noticing and Wondering. Group B spent 15-20 minutes. We then largely ran the session as I’ve detailed in the earlier blog post.

What effect do I think Notice and Wonder had? I noticed that students were keen to try the problem for themselves. They made sense of the situation, became intrigued and engaged, and then made the problem ‘theirs’. As a group, students saw that others had interesting ideas. They added on to each other’s thinking. I suspect that it also smoothed the way for working together more intensely once we broke into smaller groups where students didn’t necessarily know one another. It also became more natural for them to Notice and Wonder as the session progressed. All in all, it’s a great modification to a thoroughly engaging activity.


[1] I trialled this with teachers at the MASA conference in April.

Advertisements

Another party puzzle

More party puzzles! This one is from a thoroughly-recommended book, Puzzle Based Learning1.

Mr and Mrs Smith invited four other couples for a party. When everyone arrived, some of the people in the room shook hands with some of the others. Of course, nobody shook hands with their spouse or themselves, and nobody shook hands with the same person twice.

After that, Mr Smith asked everyone how many times they shook someone’s hand. He received different answers from everybody.

How many times did Mrs Smith shake someone’s hand?

At first glance it seems that there is not enough information to solve the puzzle—which is why I like it! Once we consider each piece of information, we can put the bits together to find a solution.

Warning: mathematical spoilers (but not the solution) ahead. I’ll post the solution in the comments.

Some prompts:

  • How many people are there at the party?
  • What is the minimum number of handshakes possible?
  • What is the maximum number of handshakes possible?
  • Can you draw a diagram to represent the handshakes made by the person who made the most handshakes?
  • What can you conclude from this?
  • What can you add to your diagram?
  • Can you now solve the puzzle?

Good luck!


[1] Michalewicz, Z., Michalewicz, M., Puzzle Based Learning, Hybrid Publishers, 2008. pp 99-102.

Ramsey’s party problem

I love games that require no special equipment because they can be played at a moment’s notice. This is one of my favourite pen-and-paper games. It is played on the complete graph K6. In other words, a board with six dots where each dot is connected to every other dot by a line.

SimBlack

Although the game-board can be drawn up for each game, I like to have pre-printed boards which can be put inside plastic sleeves for use with erasable marker pens.

To play, two players each choose a different colour. They take turns colouring any uncoloured lines between two dots. The first player to complete a triangle made solely of their colour loses.

A natural question to ask when playing games is whether or not there will always be a winner (or loser), or if the game can end in a draw. (We are also often interested in whether there is an advantage in being the first (or second) player.) Play a few times. What do you think?

What I like about this game is that the reasoning required to determine if there will always be a winner is accessible by many children and adults, but it is also the entry point into a rich area of mathematics called Ramsey theory.

Warning: the rest of this post contains mathematical spoilers.

The game described earlier is called Sim, after the mathematician Gustavus Simmons who was the first to propose it. It is also equivalent to a puzzle posed in The American Mathematical Monthly in 1958. ‘Prove that at a gathering of any six people, some three of them are either mutual acquaintances or complete strangers to each other.’ That is, we can represent the six people by six dots. We can colour the lines between two acquaintances in red and the lines between two strangers in blue. The problem now is to prove that no matter how the lines are coloured, we can not avoid producing either a red triangle (between three acquaintances) or a blue triangle (between three strangers).

So, if we can prove that we cannot avoid producing a triangle of either colour, then we have proven that the game can never end in a draw. (In fact, computer search has verified that the second player can win with perfect play.)

Consider the diagram below. There are five edges leaving node A. In a completed game, they will be coloured red or blue. At least three of them must be of the same colour, say red. Now, in triangle ABC, edge BC must be blue (else we have a red triangle). Similarly, in triangle ABD, edge BD must be blue. And, in triangle ACD, edge CD must be blue. But, we have just formed a blue triangle: BCD!

SimProof

We could have posed this question in reverse, that is, what is the minimum number of guests that must be invited to a party so that at least three guests will be acquaintances or at least three guests will be strangers.

The game of Sim shows that it is possible to invite six people to a party and have either at least three guests as acquaintances or at at least three guests as strangers. We can show that this is the minimum number of guests by considering a party with five people and showing that it is possible to have no triangle of the same colour between any three people.

Sim is a demonstration of the Ramsey number R(3,3) = 6. In the language of parties, the Ramsey number R(m,n) = p says that p is the minimum number of guests required at a party so that at least m guests are acquaintances or at least n guests are strangers.

There is a lot written about Ramsey numbers; I’ve barely scratched the surface here. (I first read about them in Martin Gardner’s Colossal Book of Mathematics.) Finding Ramsey numbers is an active area of mathematical research, partly because they are fiendishly hard to calculate. At present, only a handful are known. The last word is for Paul Erdös, the great Hungarian mathematician. (The extract is from The Australian newspaper on Wednesday 21 April 1993 when Australian mathematician Brendan McKay and American colleague Stanislaw Radziszowski found R(4,5)=25).

If an evil demon threatened to destroy the Earth in two years unless we could tell him the value of R(5,5), our correct response, said Erdös, should be to devote all mankind’s resources to the problem — we could probably solve it in two years. But if the demon instead asked us to tell him the value of R(6,6) — then, said Erdös, we should devote all our resources to finding a way to kill the demon.

Fold-and-cut

Whenever I say this, the unfinished sentence in my head is ‘Fold-and-cut, baby! Fold-and-cut.’ I am totally weird.

The fold-and-cut theorem states states that any shape with straight sides can be cut from a single sheet of paper by folding it flat, possibly with many folds, and making a single straight complete cut.

I have been impatiently waiting to try this activity with others. Last night was the fourth in a series of teacher professional development workshops loosely based around developing mathematical thinking through puzzles and games. As Adelaide edges into winter and the days get shorter, it’s getting harder to rush out of class and head to an early evening PD session instead of a glass of wine on the couch at home. The semester is drawing to a close, and energy levels are low. I’m sure others feel the way I do, so I decided to open with a ‘slow math’ activity — gentle, therapeutic folding and cutting.

This post is doubling up as the list of resources for participants. We didn’t use everything listed below, but it’s all worth checking out.

We started with the first 2:30 min of Katie Steckles’ Numberphile video on the fold-and-cut theorem.

Then I gave out copies of pages 6-8 of Joel David Hamkins‘ handout. (You can get it at this page.) Printing each page on different colour paper helped distinguish between problems — and it looks pretty! Have lots of copies for participants in case they want to try again. Last night it turned out that two per person was enough, but with students I’d suggest a few more.

At the end we watched the rest of Katie’s video, including her demonstration of cutting all 26 letters of the alphabet, each with one cut. We also had a quick look at the more intricate fold patterns on Erik Demaine‘s site. Erik also lists mathematical papers about the fold-and-cut theorem. We spent just over 30 minutes on all these activities.

I had forgotten in the moment about this suggested progression (easy to hard) from Bowman Dickson: Equilateral Triangle → Square → Isosceles Triangle → Rectangle → Regular Pentagon → Regular 5 pointed Star → Scalene Triangle → Arbitrary Quadrilateral. (Read how his fourth graders went with it here.) I did remember that a scalene triangle is supposed to be reasonably tricky, so I threw that out as a challenge. I suggest having lots of blank paper and rulers available so that people can try their own shapes.

Patrick Honner has a thorough list of folding resources, including fractal one-cuts, which I’m keen to explore further. Kate Owens has a detailed agenda for a four-hour teacher workshop on paper folding with links to standards. I noted her advice about needing strong hole punches for ‘one-punch’ activities, which is why I bypassed those particular challenges last night. Mike Lawler explored this as a Family Math project with his sons; videos and details here. Finally, the Art of Mathematics suggests it as a good ‘first day’ activity. See links at the bottom of this page of other suggestions of what to do on the first day of semester.

Have you done these activities? Do you have more resources to share? I’d love to hear about your experience.

Update (27 July 2016)

Having done this with pre-service teachers this week, I have two more resources to add:

  • An updated five-page handout. The first three pages with symmetric shapes (from JD Hamkins). The final two pages (from Patrick Honner) include irregular shapes and a Koch curve.
  • Chapter 3 of the free book ‘Art & Sculpture’ from Discovering the Art of Mathematics. This has a fabulous sequence of questions.