# Skyscrapers

This is a quick post mainly for the benefit of my ‘Developing Mathematical Thinking’ (#math1070) students.

#### Introducing the puzzle

Skyscrapers are one of my favourite logic puzzles. They are a Japanese creation, introduced at the first World Puzzle Championship1 in 1992.

Skyscrapers are a type of Latin Square puzzle. A Latin Square in an n × n grid filled with n different symbols, each occurring exactly once in each row and exactly once in each column. (Sudoku is another type of Latin Square puzzle).

In a Skyscraper puzzle the objective is to place a skyscraper in each square, with heights between 1 and n, so that no two skyscrapers in a row or column have the same height. The numbers (clues) on the outside of the grid tell us how many skyscrapers are visible from that position. (I like to imagine that I’m standing on that number and along the street that is the row/column.) Shorter skyscrapers aren’t visible behind taller ones.

We use logical deductions to solve the puzzle. For example, in the puzzle below, the clue ‘4’ tells us that the skyscrapers must appear in ascending height order in that row/column. Similarly, the clue ‘1’ tells us that the tallest skyscraper must be adjacent to the clue. That leads us to the partially-filled grid below. If you want to solve it yourself, the solution is at the bottom of this post. You can also play them online at Brain Bashers.

#### Hands-on skyscrapers

It is fairly easy to turn a skyscraper puzzle into a hands-on activity — just choose objects of different heights. Teachers often use linking cubes. You can also be more creative; at David Butler‘s One Hundred Factorial gathering at the University of Adelaide in May, we experimented with video cassettes (remember them?) and cups of different sizes.

A while back, I wanted to make several hands-on sets for 5 × 5 grids to use with groups of school students. They needed to be cheap, lightweight, compact and portable. So, I made paper cylinders that nestle inside each other. You can download and print the skyscraper cylinders. The tabs are meant to show where to overlap and tape. You can use them with these puzzles (print A3 size): Puzzle 1, Puzzle 2, Puzzle 3, Puzzle 4, Puzzle 5.

#### Skyscrapers in the classroom

My plan for MATH 1070 was curtailed by our short week (Week 3). I had planned the activity with these goals:

• Form visibly random groups with four students so that students could meet a few more classmates.
• Work collaboratively towards a common goal (and contrast this with the competitive nature of Prime Climb last week.)
• Practice claims and warrants as part of the focus on Maths Disputes: ‘I think <claim: this number goes here> because <warrant: my reason>.’

There are a variety of reasons to use skyscrapers in the classroom; you might like to read these posts by teachers: Mary BourassaMark Chubb, Sarah Carter. Any activity introduced into the classroom should be intentional. You might like to think about these dot points. Mark has a fuller list in his blog post.

• If giving these puzzles to individual students is different than to groups of students.
• If a physical model is different than a pen-and-paper version.
• If you’ll use it as part of a lesson or as a ‘time filler’.
• What you’ll do if students give up easily.

If you give them a go, let me know what you think!

# Redux: #NoticeWonder and #PrimeClimb

Last year I wrote a post about using the two simple questions ‘What do you notice?’ and ‘What do you wonder?’ with my maths pre-service teachers to dig into the mathematically-rich image that accompanies Dan Finkel‘s game, Prime Climb.

This year, I wanted to turn this into a student-driven rather than teacher-led activity. I also wanted to create opportunities for even deeper mathematical exploration. In this post I’ll briefly outline how this unfolded.

#### Using Tiny Polka Dots for visible random groupings

A goal for this year is to do better at helping students be good mathematical colleagues. I’ve been making heavy use of Chapter 12 of Tracy Zager’s book to guide this endeavour. I also wanted to begin using visibly random groups to build our mathematical community. However, I’m mindful it can be confronting to work with complete strangers, so I had students pair up first with someone they knew. (Pairing students up was also to serve another purpose that will become apparent later.)

I randomised the pairs by handing out cards from another of Dan’s games, the delightful Tiny Polka Dots. I deliberately used cards with different representations of 1, 2, 7 and 9 (with 8 as a back-up). Pairs with a different representation of the same number formed a group, one at each table. I gave them the rest of the cards for their number, and we did a quick ‘Notice and Wonder’ on the different representations.

#### Different-sized grids for Prime Climb

Rather than show students the Prime Climb hundreds chart arranged in columns of 10 (the image earlier in this post), I wanted to foster noticing and wondering by having them construct the charts themselves with mini-cards of the numbers: physically handling, examining, ordering and organising. The (roughly) six students at each table arranged the cards into charts with the number of columns corresponding to their Tiny Polka Dots card (with 1 and 2 corresponding to 11 and 12, respectively). Click on the images below to make them bigger. What do you notice? What do you wonder?

To add impetus to the discussion, I relayed that part of the weekly task (contributing towards their course grade) was to individually write a forum post with at least five things that they noticed and wondered about their charts. It was heartening to see students collaboratively generating lists of their many observations.

#### Playing Prime Climb to ‘make thinking visible’

The final part of this activity was to have students play Prime Climb in a way that compels them to articulate their thinking. One observation from playing Prime Climb at One Hundred Factorial with David Butler was that playing in pairs had an initially unexpected benefit of making thinking visible. (Side note: Bodyscale Prime Climb—where the numbers are A4 sized and the player is the pawn—is the most wonderful way to experience this game. Walking the board gives a different perspective to the relationship between numbers. You also need at times to shout your thinking across the board to your partner, which really makes thinking visible! Read more in David’s blog post.) We used a modification of the rules devised by David. This is the printable version that I wrote based on the rules that come with the game.

Collaboration and competition reached ear-splitting levels, so much so that we were asked to quieten down from those in the classrooms around us. (I sheepishly and profusely apologised later.) This was definitely a good start to building community.

#### Further exploration

While the in-class activities concluded with playing Prime Climb, there is so much more exploration provided by the cards. David, and later I, explored the different patterns that emerge when the cards are arranged in different ways. Others chimed in on Twitter with ideas; click the links above or look at the images below for more. The Prime Climb colouring is such a rich medium!

# #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.

# Counting in unexpected ways

It was a delight to recently spend five days working with students and teachers in Alice Springs at the invitation of MTANT, the Mathematics Teachers Association of the Northern Territory. I then spent a week in bed with the flu, which is one reason I’ve recently lost my voice (both physically and online).

The main purpose of the visit was to join the 8th Annual Maths Enrichment Camp at Ti Tree School, in a small remote town 200 km north of Alice Springs. Students travelled from all over Central Australia, some from as far as the Aboriginal community of Hermannsburg, 320 km south (that’s a short distance in the Northern Territory!). The camp runs Friday night to Sunday morning, and is full of fun activities (mathematical and non-mathematical) for kids, and teachers, to engage with. This was my first Maths Camp, and I was thrilled to be invited; thanks @matt_skoss!

This year the Ti Tree Maths Camp attracted around 35 students from Years 4 to 10. Students were divided into three groups and on Saturday rotated through four activities, called ‘Worlds’. Thus, these activities needed to accommodate a broad range of mathematical expertise. To add extra challenge, I rarely work with students in the lower years, so I relied on a couple of trusted friends to help determine whether my planned modifications would be appropriate.

In this post I briefly describe how younger students responded to two of my favourite activities, which I’ve previously written about: The Game of SET and Domino Circles. I doubt that this is going to be revelationary to most teachers, but I am always learning how students make sense of mathematics (younger students, in particular), so I want to record my observations for the future.

### Counting Dominoes

I worked on this problem with a combined group of Years 5 and 6 girls from Bradshaw Primary School and Araluen Christian College in Alice Springs.

Display this image. What do you notice? What do you wonder?

Responses include:

• I notice: that there different numbers of dots on a domino.
• I notice: that there are two groups of dots on each domino.
• I notice: that the dots are different colours.
• I wonder: what is the highest number of dots on a domino? (A fascinating side discussion commenced as we had to resolve whether we meant in total or on one half of the domino. We decided that in a Double 6 set, the highest number is six. What do you notice and wonder now?)
• I wonder: what is the lowest number of dots on a domino? (Zero.)
• I wonder: how many dominoes are there in the set? (My response is usually ‘Good question! I wonder if we can work that out?’ :))
• I wonder: can a domino have more than one instance of the same number of dots? (Yes — I show a ‘double domino’, like 2|2.)
• I wonder: is there exactly one of every combination of numbers of dots? (Yes.)

Usually my next question is to ask students to calculate how many dominoes there will be in a set. Some students start by drawing them all out. For this students I might show an image of a Double 18 set—too many to draw, right? This encourages students to find, and then explain, a formula for the number of dominoes in a Double ‘n’ set.

However, for younger students that I hadn’t met before, I was concerned that this question might be too challenging. Instead, I handed out sets of dominoes and asked students to have a look at them. Then I revealed that each set was missing a domino. Could they work out which domino was missing?

As you might expect, students needed to find a way to organise their dominoes so that they could identify the missing one. Several groups made arrangements like this. It’s easy to spot the missing domino now, right? How else could you have arranged the dominoes to make this discovery?

I gave one large group two sets of dominoes (one paper, one physical) in case they wanted to work in smaller groups. I was delighted to find that they instead used both sets of dominoes in tandem. It looked something like I’ve reproduced in the photo below. Can you spot the missing domino from each set?

There was an interesting moment in the middle of this activity as we discovered that some sets had more than one domino missing, and some sets had duplicates. (Guess who didn’t double-check the domino sets before starting the activity? <blush>) This could have been a disaster, but I took it as a true problem-solving experience for the group. We sorted out our sets eventually!

The rest of the session was largely spent exploring this question:

Is it possible to arrange an entire set of dominoes in a circle so that touching dominoes have adjacent squares with identical numbers?

Once you’ve experimented with a set of dominoes in which the highest number is six, explore whether it is possible for sets of dominoes where the highest number is different.

We finished with a quick ‘Notice and Wonder’ with this short promotional video by Cadbury, in which they set up blocks of chocolate in a suburban street, and knocked them over like dominoes. I wish I could remember all the rich wonderings the students had — they were fabulous!

### Counting SET cards

My chosen activity for my ‘World’ at the Ti Tree Maths Camp was The Game of SET. I’ll briefly recap the game, before talking about how students counted their set cards.

SET is a card game. Three of the cards are shown below. What do you notice?

Students eventually identify a number of attributes of the cards. Sometimes (but not often) they generate more than we need for the game. I acknowledge them and ask if we can focus on four particular attributes: number, shape, colour and shading. We notice that each attribute has three different values that it can take. For example, shape can be ‘oval’, diamond’ or ‘the squiggly thing’.

I confirm that these are all the possibilities of values of attributes. When I work with students in higher grades, my next question is usually as follows.

If a SET deck contains exactly one card of every possible combination of attributes, how many cards are in a deck?

To adapt this question to lower years, I did something similar to what I did with counting dominoes. But instead of removing a card, I asked them to find a way to be sure that they had exactly one card for every possible combination of attributes in their deck.

Their natural ability to group by features that were the same, and to organise in a systematic way, was not unexpected. But I enjoyed seeing the varied ways they went about this.

For example, these two girls made three rows (shown vertically in the photo). Each row corresponds to a colour. Within each row, they grouped first by shape. For example, all the red diamonds, then all the red squiggles, then all the red ovals. Within each shape they grouped by shading. Within each shading they organised by number. One explanation was that, for a particular colour, they knew that there were nine cards for a particular shape. There were three different shapes. So there were 27 cards in one row. There were three rows of different colours. So there were 27 × 3 = 81 cards.

Some students started grouping by colour, but in a different way. In this grid, each row corresponds to one particular shape. Each column corresponds to one particular shading. Each ‘entry’ in the grid contains three cards, grouped by number. There were two other grids like this, each for a particular colour. One explanation was that, for a particular colour, we get a 3 × 3 grid where each entry contains 3 cards. So, each grid has 27 cards. There are three different grids, each corresponding to a different colour. So there are 27 × 3 = 81 cards.

There are four different ways of organising shown in the photo below. In the left bottom half, a student is organising in a way similar to the grid method. Focus on the larger cards in the far right. These girls have nine columns. Each column corresponds to a particular colour and shading combination. For example, the far left column are cards that have purple shapes that are completely filled in. Within each column, they organised the cards in groups of three. The three groups are organised first by shape. Within each group the cards are organised by number. Their explanation is that there are nine columns, each with nine cards. So they have 9 × 9 = 81 cards.

I loved all these ways — and more not described — that students found to count the number of cards they had in their deck. After students had completed their work, we congregated together and went on a tour of the room. Each smaller group explained to the whole group how they had organised their cards and confirmed that there were 81 cards in the deck.

An unexpected advantage of this approach is that students discovered for themselves how to make a SET, because of the natural ways that they grouped cards. In the game, a SET is a group of three cards where, for each of the four attributes, the features are the same across all three cards or different across all three cards. For example, the three cards below are a SET because shape is all-same, fill is all-same, number is all-different, colour is all-different.

Once students understood how to make a SET, we made a new discovery within their work. Consider again the top cards of the 3 × 3 grid shown below. Each row, column and diagonal forms a SET. It’s like a magic square. A magic SET square.

Meanwhile, the students who organised their cards into a 9 × 9 grid decided to keep their columns the same, but rearrange it so that each row corresponded to a particular number and shading combination. I’ve reproduced it below. With a little bit of prompting from me, they discovered that they had a kind of super magic SET square. Can you see what I mean? So cool!

The rest of the session was spent playing the game, and talking over some of our SET-finding and problem-solving strategies. A rough description is here.

What struck me is that these students’ understanding of how to form a SET was much more solid, and developed so much quicker, than many other older students I work with. This is because usually I explain, rather than have them explore. At Maths Camp I was reminded — again — that even in an activity full of moments for discovery, there are still more opportunities to slow down and let students construct knowledge for themselves.

# 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.

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!

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.

# The Joy of SET

Just over a week ago I shared one of my favourite mathematical puzzles. Today I’m sharing my favourite mathematical game, SET. There is something in this game for young children to mathematics professors. I give SET workshops each year, for Year 8 students up. My slides are here, along with some notes I wrote several years ago. I suggest at least 90 minutes. A brief outline, along with some interesting questions, is below.

SET is a card game. There are four attributes of a SET card. The three cards below show all possible values of the attributes.

If a SET deck contains exactly one card of every possible combination of attributes, how many cards are in a deck?

A SET is a group of three cards where, for each of the four attributes, the features are the same across all three cards or different across all three cards. The three cards below are a SET because shape is all-same, fill is all-same, number is all-different, colour is all-different.

If two cards are selected at random from the deck, how many cards, if any, can be paired with the first two to complete the SET?

How many SETs are possible (including those where a card is used in more than one SET) among all cards in the deck?

To play SET with others, lay twelve cards out on the table. The first person to find a SET calls ‘SET’, shows it to the rest of the players. The empty spaces are then replenished from the deck. If the players agree that there is no SET on the table, three more cards are dealt. At the end of the game, the person with the most SETs wins.

Many interesting questions arise when playing a game of SET, for example:

1. Which types of SETs are most likely in a deck? (And how can you use this to your advantage when playing?)
2. Can only three cards be left at the end of the game?
3. How many cards is it possible for there to be on the table with no SETs present?

There are more interesting questions at the end of my slides, in an article1 from the Mathematics Teacher, and in multiple resources on the web. Constructing magic squares with SET cards is also fun!

To investigate the types of SETs most likely in a deck, I suggest letting students first play a game and then have them analyse how many of the different SET types they found (for example, three attributes all-same, one attribute all-different; two all-same, two different). Allow at least 30 minutes for this. Collating and comparing results between groups of students leads to a rich discussion.

Mathematicians have used the theory of vector spaces to prove that it is possible to find 20 cards without a SET — termed a maximal cap for four attributes. Exhaustive computer search also shows that every collection of 21 cards contains a SET. But the vector space theory isn’t very accessible for most. I’m interested in what progress someone with a few problem-solving skills but no advanced mathematics can discover.

For three attributes, it is possible for students with some careful reasoning to find the maximal cap of nine cards without a SET. Four attributes is more difficult; I’ve not yet managed it. Yet, I’ve had three students (one high-school, two pre-service maths teachers) prepared to tackle this project. Hearing about strategies they’ve devised — and the patterns they’ve discovered that come only from deep thinking and research on a problem — has possibly been the most fascinating part of this game.

At same stage I’ll tell you how to play Quarto with SET cards — but that’s another post.

Have you or your students played SET? How have you introduced it into the classroom? What mathematical questions do you and your students have? I’d love to extend my investigations into this game by hearing about your ideas.

[1] Anne Larson Quinn, Robert M. Koca Jr, Frederick Weening, ‘Developing Mathematical Reasoning Using Attribute Games’, Mathematics Teacher, 92 (9), 1999, pg 768-775.