New TYPES of puzzles

We're over halfway done with the semester and my Games and Puzzles class has been fascinating and rewarding in many ways.

As I mentioned last time, our first foray in pencil puzzles was into Battleships, after which we've explored the pencil puzzles Nurikabe, Skyscrapers, and Masyu for a few weeks each. I was starting to become wary of the structure rut that I had been finding myself in, in which we would start with a new puzzle, investigate strategies in detail, try creating a puzzle of that type, and then the final result of the unit was the creation of a puzzle of that type given some twist.

So in class on April 6, I suggested the following challenge: work in groups of 3 or 4 to create a hybrid puzzle incorporating concepts from at least two of the four puzzles we have mastered this semester. I was encouraged by how quickly they took to the task. The discussions remained focused on the challenge and the discussions included all members of the group. In my other project-based classes I have a tendency to be very hands on. But for this challenge the students didn't need me at all---which is great! The ideas are flowing and the students are pulling in all the expertise that they have accumulated throughout the semester. I stepped in a few times to diffuse some concerns such as what is allowed in this new type of puzzle (Anything!) or when a group was having a heated discussion about the type of clues that they should include (What if you created two different types of puzzles?!)

My students started the semester solving the logic puzzles haphazardly and moved to thinking critically about solving strategies. They have progressed from creating puzzles of a specified type often without regard to uniqueness, to thinking critically about creating puzzles using a puzzle solver's eye. Now they are creating new types of puzzles! These transitions have been gradual and I can see my students' confidence building.

Announcing the Circular Teaching Squad

Abstract: I discuss the rationale for changing the name of our lunch discussion group. Join us March 22nd!

For a couple years now I have been moderating lunch discussions about teaching and learning on campus here at Queens College.  I had been doing so under the guise of the Experiential Learning Group started back in 2012.  After our last lunch, my friend and kindred spirit striving for excellence in teaching Nathalia Holtzman came up to me and suggested that I move away from its association with the defunct Experiential Learning group.

I was especially swayed by the argument that the discussions we have been having about teaching and learning are much broader than "experiential learning", and perhaps that misnomer was keeping other faculty away who might want to participate in a more general discussion about teaching and learning but were either less interested in experiential learning, or unsure of what this buzzword entails.  This set in motion the idea to change the name of the group.

The name I chose for the group is the Circular Teaching Squad.  I like this name for multiple reasons.  First, it is unusual, and would make people pause, question, and consider it, if only for a small while.  Second, it is playful, which is a quality I very much enjoy.  Third, it can be easily shortened to "The Teaching Squad" or just "The Squad".  Fourth, isn't SQUAD an amazing word?  It even has a Q in it, just like Queens!  Fifth, it conveys the idea of a "Teaching Circle", where we share ideas as equals.

Last, what truly inspired me to choose the name is the concept of a Circular Firing Squad, in which a group of people (often political candidates) attack each other, often weakening themselves so much that they lose the bigger fight.  Instead of weakening each other, our discussions involve shooting ideas about teaching and learning into a circle of peers who teach diverse subjects, transmitting practical knowledge and strengthening our bonds as educators at this public university.

The Circular Teaching Squad will meet from 12-2 on Tuesday, March 22 in the President's Conference Room #1 in the Queens College Library.  (Stay the whole time or drop in informally!)  The topic of discussion will be about the ways in which we teach students how to learn.  Should we teach these ideas in a more explicit way?  In what ways do we teach students how to take ownership of their learning? While the is the first meeting under the new name, it will continue our tradition of invigorating the discussion of teaching and learning on campus!  If you're in the neighborhood, we'd love to see you there!

This blog post is part of the Queens College Teaching Circle blog; it is cross-posted on my personal teaching blog, Math Razzle.

Student Forays into Pencil Puzzles

Abstract: In the first six days of my pencil puzzles class, my students have had to think deeply about solving and creating Battleships puzzles. This post discusses the day-to-day class mechanics and assignments.

Now I'll talk about the first few days of my pencil puzzles class.  (Previous post setting the stage.)  I started the first day of class by qualifying that this class was new to me and would be an adventure for them and for me.  I explained that there wouldn't be any lectures, there wouldn't be any exams, and that it was unclear exactly what the assessments would be---the initial ideas I included on the syllabus were:

• One-page-long essays about the history of certain puzzles and related solving techniques.
• Curation of content for a wiki about puzzles, puzzle solving, and puzzle creation.
• New puzzles with accompanying discussion of the methods used to create them.
• The uploading of puzzles to web repositories.
• The creation of a puzzle trove.

I conveyed that I wanted the class to be a student-driven discussion about the solving and creation of pencil puzzles.  The rest of the first day was filled with a worksheet with four types of pencil puzzles and the students worked on two of the types.  The first homework assignment was to work on 12 Battleships puzzles I had copied from puzzle magazines I had laying around and for the students to address the following prompts:

• What strategies are you using? (Give them names. Give a simplified example of what the strategy says. Only including the relevant parts of the puzzle board.)
• What makes one puzzle harder than another? Try to find a way to explain it in words.
• How does knowing that there is one unique solution help you?
• Highlight areas where you get "stuck". If you are able to get past being stuck, how were you able to do so? Take a picture with phone/camera. What happened at that moment in time? Why are you able to make a mark? What logical reasoning are you using? What was the "AHA" moment?

On the second day of class, students discussed their strategies and I served as scribe.  In our brainstorming session before the start of the semester, Sarah Mason had drawn on her own classroom experience to suggest that we give the solving strategies names so that they would be easier to discuss, similar to how names are given to strategies on Sudoku solving websites.  I thought that this was a great idea, since it would solidify a strategy with a clever name and as mathematicians, it is always good to define commonly occurring concepts.  However, the naming of techniques didn't take hold.  Either there was a lack of creativity for punny names, or just no appetite for giving the techniques names, so I abandoned this idea after the second day of class.

I found it especially worthwhile when a student projected onto the screen a before-and-after pair of pictures.  He showed a situation where he was stuck, and then detailed the logical reasoning that it took to get to write the next mark on the paper.  This sort of play-by-play analysis is crucial in conveying one's logic and strategies to the class.  The detailed discussion of strategies we had in class was universally considered to be helpful, and led to the next homework assignment of completing more complicated battleships puzzles before the next class.

On the third day we spent the whole day working through the following battleships puzzle from start to end because every single student was stuck:

This led to an in-depth discussion about the concept of a decision tree, and that battleships puzzles that are hard often have a wide and deep decision tree.  In other words, in order to figure out the next mark to place, there are many possibilities for the placement of the next ship (Either here or here or here or ...) and for each of those choices, the sequence of logical steps leading to a contradiction is complicated.  Our discussion about this puzzle was especially interesting because of the appearance of the 21112 clues on the bottom left.  We discussed how the uniqueness of a solution implies that there must be some symmetry in the placement of ships, restricting the possible next moves.  The homework for the next day was as follows:

• Create two or more Battleships puzzles with some intrigue, at least one of which is a Junior Battleships puzzle. As you do so, answer the following questions.
• What is hard about creating a puzzle?
• How do you know your puzzle has a unique solution?
• How does creating puzzles influence how you solve them?

So on the fourth day, the students brought in their hand-crafted battleships puzzles, we arranged the chairs in a circle and had a deep discussion about the difficulties involved in their creation.  As I moderated this lively discussion about puzzle creation, I transcribed a number of themes that kept reappearing, from which the homework assignment for the following day emerged organically:

• Create one or more Battleships puzzles that push the boundaries imposed by the normal structure of a Battleships puzzle.
• Write up a page of discussion about the making of your puzzle.
  1. What conscious decisions went into its construction? What boundaries did you push? Which decisions make the puzzle easier or harder to create? Easier or harder to solve?
  2. How does puzzle uniqueness play a role in its construction?
  3. How does the tension between "uniqueness of solution" and "puzzle difficulty" come into play?
  4. What direction were you using in its construction? Forward? Backward?

The last half-hour of this fourth day was spent working on peer-created puzzles that had collected copied in our office.  One issue that arose is that some of the puzzles did not have a unique solution.  This served as a warning that everyone needs to be very careful when creating the puzzles.

The fifth day was a day of presentations.  Each student came to the front of the class and projected their puzzle on the screen, and talked for a few minutes about their puzzle and the boundaries they pushed when creating it.  Some students changed the shape of the board.  Some students changed the instructions.  Some students changed the number and types of ships.  One student even created a battleships puzzle on a torus!  To me the most amazing part of the day was that the nine students who presented had each pushed the boundaries in distinct ways!  We ended the day by introducing our next type of pencil puzzle: Nurikabe (ni, cp, pp), which the students would have to practice solving for the next class.  The following screenshot about Nurikabe is from Conceptis Puzzles:

Monday 2.22 was the sixth day of class, and students came in to discuss their trials and tribulations in solving Nurikabe puzzles.  We once again arranged the chairs in a circle, and I moderated a discussion about the difference between Battleships and Nurikabe.  What was the most fascinating to me was the palpable interest in and anxiety about creating Nurikabe puzzles, not the strategies involved in solving the puzzles.  One student explained that this feeling was based on the fact that at some level she was comfortable solving puzzles, but because this class was the first time that she had to create puzzles, that is where her mind went first.

So far, I am very pleased by how well this class is going.  Every student is participating in class in productive discussions.  The students are excited about and engaged with the material.  I am a bit concerned about the class becoming tedious for the students; I don't especially want the semester to fall into a rut of "introduce a new type of puzzle---learn how to solve it---learn how to create it".  If you have any suggestions about what else might break up the monotony and let the students get more out of the class, I'd love to hear your thoughts!

This blog post is part of the Queens College Teaching Circle blog; it is cross-posted on my personal teaching blog, Math Razzle.

Teaching Pencil Puzzles

This semester I am teaching Math 555: Games and Puzzles for the first time.  The class does not serve as a prerequisite for any other class and it does not fulfill any degree requirements more than miscellaneous math credits, which means that I have a free hand to teach the content I want to teach.  This class is normally offered during the summer and recently the class has been taught with a focus on game theory.  Instead, I decided to focus on one of my favorite past times, pencil puzzles.  These are small puzzles that can be solved using pure logic.  Many people will have tried to solve the most ubiquitous of pencil puzzles---Sudoku!

(A quick tangent to provide good pencil puzzle links!  Three high-quality websites for downloads and online play of pencil puzzles are Nikoli, Conceptis Puzzles, and Puzzle Picnic (user-generated puzzles).  Two excellent puzzle magazines (which also include word puzzles) are Games World of Puzzles and Dell Math & Logic Problems.)

I wanted to get a bit off the beaten path and away from Sudoku, and not even focus on the two types of pencil puzzles that I suggest to friends (and enemies) as being "after Sudoku": Hashi (ni, cp) and Slitherlink (ni, cp, pp).  In fact, a couple students had already seen these puzzles.  Instead, I decided to first start with Battleships (cp, pp).  This is a solitaire version of the two-player game you may have played pictured here:

In both games, you are searching for a fleet of ships that are placed in a grid.  In the two-player game, your opponent places ships in a way that they think will be difficult to find, and you guess the positions.  In the pencil puzzle solitaire version, a fleet of ships is hidden and your only clues are the number of grid cells in each column and in each row that are covered by some ship.  With these numbers, there is one unique solution that can be found purely by logic.  Here is a moderate-to-difficult puzzle created by user anurag.sahay on Puzzle Picnic:

There is the additional rule that no two ships can touch, even diagonally, which reduces the amount of possible placements.  Try out a first few sample puzzles at Conceptis Puzzles.

This was the first class I have taught in which I have no idea how the semester is going to play out---I am taking the class day by day with the ultimate goal of compiling a collection of student-created puzzles of many types.  In my next post, I will discuss the first days of the semester and how things have gone.

A quick thank you to Sarah Mason, who acted as a sounding board for the course structure before the beginning of the semester, to my students for being enthusiastic participants on this adventure, and to my department for their support in allowing me to teach such an amazing class!

This blog post is part of the Queens College Teaching Circle blog; it is cross-posted on my personal teaching blog, Math Razzle.