Tag: puzzle

Building Strategic Thinking with Dominoes

My 5th grade gifted class revisited the game of Dominoes last week. It took some review, but they enjoyed playing the game. I told them that one of the reasons I had taught them the game was because it is a classic that they could play with grandparents and other elderly people, bridging the gap between generations. The game has been in existence for over 900 years! 

In addition to the game being old, it also presents an opportunity to practice strategic thinking. In an effort to prove this to my 5th graders, I have begun dreaming up scenarios where a player might use analytic skills to make a counter-intuitive move that would benefit them in the long run. 

There are times during a game when you have more than one Bone (Domino) that you can play, but none of the plays will give you points. Sometimes, it does not matter which one you put down, but other times you can plan ahead. Much like you would in chess, you can set up future moves by arranging the Bones to meet your needs. Playing them in a particular order would benefit you more. 

I planned on showing my 5th graders what I meant by setting up scenarios of games and taking pictures. I have done that many times to teach the problem-solving aspect of Dominoes. 

Using photos as teaching tools works great on interactive devices.

Then I thought, Why not have my gifted students make up the puzzles themselves? I will give them the parameters, and they have to try to figure out how to show the need for strategic thinking through constructing an image of a hypothetical game. 

The puzzle would be an image showing Bones (Dominoes) already played, Bones available to a player (standing up so Pips or dots were showing), blank sides of the opponent’s Bones, and maybe a Boneyard (unused Dominoes). 

If you are a novice Dominoes player, some of this vocabulary might be new to you. Bones are the game pieces, named after what they were originally made out of; Ivory or elephant tusks (bones). The Boneyard is made up of the unused Bones lying face down. Face down means that the Pips or dots on the bones are not showing. All you can see is a blank Bone or the uniform design that is printed/carved on every one of the 28 Bones of the set. Bones often have something decorative on the side without Pips, so that players can identify the 0-0 Bone more easily. Every Bone has two numbers on it. There are two ends of the number side of a Bone. No two Bones have the same combination of numbers. Beginning at 0-0, the Bones go up to 6-6. 

The Plan: In order to demonstrate strategies for play, I am going to have my 5th graders come up with puzzles that point to weighted plays. In other words there will be better moves than others. People trying to solve the puzzles will have to analyze the potential moves. Which one is better and why? Puzzle-solvers will be required to explain the move they chose. 

This is from the beginning of the year.

The Work: Arrange Bones as though they had been played in a game. This means matching the ends of Bones; Six is connected to six, three to three, etc. There ought to be four lines of play that a player can connect a Bone to.

Each player has Bones left to play. One set of Bones is standing up, with the number of Pips showing. These are the Bones that the puzzle-solver has to work with. (Normally, when I am teaching Dominoes to students, I have them lay all of the Bones down, so that every student can see all of the Pips. This is so that every single play is a lesson on problem-solving. When one plays a real game, you do not show your Bones to your opponent.)

The Bones that the puzzle-solver has to work with (the ones showing Pips in the image) should have numbers that can be played. They contain the number that is present at the ends of the lines of play. One of the Bones that can be played would cause the sum of all four ends of the lines of play to add up to a multiple of five, which is how one acquires points in Dominoes. This would seem like the best choice to complete the puzzle. 

Because we want this to be a puzzle that causes Domino players to grow in their understanding of the game and not just an illustration modeling how to play, we aren’t going to make the correct answer to our puzzle be an obvious choice. A good head-scratcher will require a player to look beyond the obvious play. 

If four Bones with the same number have already been played, and the puzzle-solver has two of the remaining Bones with that same number, how likely is it that the opponent of the puzzle-solver has any Bones with that number? 

Here is your task: Make it so that playing the Bone that does NOT create a multiple of five is the better play. 

I’ll have to share this lesson with my 4th graders, as well!

How could this happen? If the opponent of the puzzle-solver is forced to draw a Bone from the Boneyard, rather than playing a Bone, not only will they not earn any points, but they will be growing the number of points that the puzzle-solver will get at the end of the round; The round that the puzzle-solver is now more likely to win because they have fewer Bones left than their opponent. 

At the end of each round the player who uses up all of their Bones first gets points from the Pips that are on their opponent’s remaining Bones. In order for the play that did not make a multiple of five in the first place (at the beginning of the puzzle-solving exercise) to be the better play, the final play must provide more points than the potential multiple of five. 

If the multiple of five would have been fifteen, and there is no way, given the Bones that are left, for the puzzle-solver’s opponent to have a total of Pips greater than fifteen (you always round up, so sixteen would go up to twenty), then not playing the multiple of five during play would not necessarily be a winning strategy. Typically, you would play the multiple of five, get the points, and hope for the best. This exercise is designed to show my 5th graders that if you plan ahead, the delay of point acquisition could very well bring a windfall of greater point tallies. Not only is this a good life lesson, but it can help them play the game better in the future. 

Now, if you want to try to figure out how to create a puzzle that fulfills these requirements on your own, without any help, go for it. You can return to this writing when/if you get stuck and need some guidance. The next section provides some helpful hints. 

If you aren’t sure where to start, or you have hit a mental block, check out these ideas.

Some Helpful Hints: 

Limit the available Bones. You can do this several different ways. One is to only give the puzzle-solver two bones to choose from. 

Another way to limit the potential outcomes is to make the lines of play long. Have most of the Bones from the set showing in the lines of play, so that the potential Bones of the opponent is narrowed to only a few possible numbers. The puzzle-solver can reverse-engineer the game to figure out what Bones are left to be played. It’s like “card-counting,” but legal;)

A very effective strategy for creating a doable puzzle is to limit the numbers in play. Idea: Make the ends of the line of play all the same number, and the puzzle-solver has the remaining Bones that contain that number. For example, there is a one at the end of all four lines of play. There are only seven Bones that have a one in them. If four of these are played, and the puzzle-solver has the remaining three, then the opponent cannot possibly play any of their Bones. 

But, the puzzle has the puzzle-solver making the next play. How can the puzzle-solver cause their opponent to have to draw from the Boneyard? See if you can figure it out.

There are a couple of ways to solve this problem. One answer is to provide the puzzle-solver with a double. A double has the same number on both sides. When this is played at the end of a line of play, it keeps that number going! 

Another solution requires more work, and could therefore be trickier for the puzzle-solver to find. Make it so that all of the Bones that the puzzle-solver possesses have numbers on them that can’t be played. You have to position every bone that has any of the other numbers on them within the lines of play. No need to worry about your puzzle-solver using up their Bones because every one of theirs contains the same number as the ends of the lines of play. 

Stack the Pips. Create lines of play that have low numbers, thus ensuring that the Bones that the opponent possesses are more likely to have higher Pip counts. In this way, even if the puzzle-solver would make a fifteen or twenty with the false-solution-Bone (the one that would make a multiple of five and seems to be the better choice for the puzzle-solver to choose), the total Pips that the opponent would have must be greater than the multiple of five. This number work is truly statistical thinking. Out of all of the Bones still available, how likely is it for the opponent to have a high enough number of Pips for the counterintuitive play to benefit the puzzle-solver more? 

This puzzle would allow for the opponent to make a play or two before the puzzle-solver is out of Bones. My student would have to work through all of the possible outcomes to ensure that the puzzle-solver would come out on top. 

3rd graders learn to play Dominoes

Try it out, and make the puzzle fool-proof. When making the puzzle, turn all of the Bones over so that the Pips are showing. Create a model of lines of play. Give the puzzle-solver the Bones they will work with. Now, look at the Bones that the opponent could have. Adjust the lines of play, so that there is no possible way for the opponent to have a way of winning. You also have to double-check that there are only Bones that would cause the opponent to have more Pips than the false-solution. Then turn over the Bones that form the Boneyard, and stand up a couple that represent the unknown opponent’s Bones. 

Conclusion:

Normally, I will do a lesson like this, and then write a blog about it. This is different. I have used my writing to think through what I want to have my 5th graders do. 

My aim is to have them build their understanding of the game of Dominoes and learn statistical analysis through the process of constructing their own puzzles, rather than just solving mine. Hopefully it will be successful, and I can write a follow up blog about how wonderful it went… or the lessons I learned through its execution, pun intended;) 

If you try this idea or one like it, please share your results. I’d love to learn feedback and improve future teaching. 

Sources:

Marcus, M. (2020). How to Play Dominoes . Cool Math Games. https://www.coolmathgames.com/blog/how-to-play-dominoes

Puzzling a Positive Attitude of Acceptance

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This year’s puzzle, “Cats” (2019)

I grew up working on puzzles with my family during the winter holiday breaks. Every year between Christmas and New Years, every member of my family could be found hunched over a temporary card table set up in the middle of the living room. Never-mind Santa Claus; The main source of magic in those days was waking up to find a gigantic portion of the puzzle completed. Sometimes, feelings of gratitude flooded the soul, since a piece that had eluded me was found and locked in place. Other times, I would feel cheated, because a large portion of the puzzle was no longer available for construction. 

The purchaser of puzzles was my mom, and she wasn’t very picky. She brought home pictures that she liked. They were typical interlocking jigsaw puzzles consisting of anywhere between one and two thousand pieces. 

Mom was the best at finding matching pieces. I remember puzzling over a section for a long time, only to have mom visit the project, survey the available pieces, and pick the most random-looking one out of the mix. Not every time, but often, she’d have flawlessly, and seemingly effortlessly found the missing piece!

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Dawning her new fuzzy, pink coat, Scarlet connects pieces that have words on them.

Now, I am teaching my daughter Scarlet how to assemble puzzles. My wife and I have given Scarlet puzzles for years, but this Christmas is the first time that she is really engaged in puzzling through a 750-piecer, independently. I came downstairs yesterday morning to find her hunched over the colorful enigma, piece in hand. As I began explaining that sometimes it isn’t enough to look at the colors of the pieces; “You have to examine the negative space to find a matching piece;” I thought about the ways people interact with one another. 

When Scarlet and I first dumped out the 750 pieces, I instructed her to turn all of the pieces over, so that the glossy, colored sides were facing up. As we did this, we placed any pieces that had straight edges into a separate pile: Those were the edge pieces. We also began forming a few other distinct piles of pieces during the flip over process. Any pieces with blue on them went into a pile separate from the pile of red pieces. 

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Puzzles prepare people for finding commonalities.

Anyone familiar with putting together puzzles knows that you first assemble the border, so that you have a literal frame. This also provides a frame of reference for where to place increasingly connected groups of pieces. Sometimes you have to open up the border to allow large sections of connected pieces to slide into the middle of it. Other times, you can pick up a handful of attached pieces and drop them into their spot. Then there are times when you work on bringing individual pieces into the frame to fill gaps and complete sections, all the while you are hunting relationships between interlocking, individual, man-made masses of color and shape that are meant to be together. 

Scarlet, Mommy, and I are only about a quarter done assembling our family puzzle, when it hits me: What if people had the same attitude about other humans that a person putting together a puzzle possesses? The whole time you work on a puzzle, you are only thinking, where does this piece in my hand belong? Who does it connect to? What spot in the puzzle is its home? How does it relate to the rest of the puzzle?

The puzzle that we are working on is tricky, in that the pieces seem to be cut right on the cusp of varying colors. I am working on assembling a section full of poinsettias and looking for an adjacent piece. I first try pieces that look similar to those adjacent the vacancy. None of those fit. Next, I examine the opening that the missing piece forms; the negative space. Are there any pieces in the pile of red that possess a similar shape? Eventually, I go ahead and try sticking red pieces into the gap, just in case there is a geometric abnormality that I am not seeing. Perhaps one will work, even though it does not look like it. That’s happened before. No success. I widen my scope of potential pieces to include ones that share the colors of surrounding pieces; greens and browns. When finally I do find the piece, I notice that the cut followed the line of red color, seeming to purposefully fool anyone relying on that. 

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The important thing to understand is that the entire time you are putting together a puzzle, you are being inclusive and accepting.

This story goes to show that while it may be advantageous to categorize puzzle pieces into separate colors, initially, this reliance on related pigments can eventually cause frustration. I would have found my missing piece faster, had I looked at the picture of the puzzle on the box, and included any piece that had red AND leaf colors/designs in my pile of poinsettia pieces. And, really, if you want to get really figurative and technical, poinsettia petals actually are leaves that turn red! 

The important thing to understand is that the entire time you are putting together a puzzle, you are being inclusive and accepting. Every single one of the 750 pieces is a potential fit for any opening. They all belong somewhere. I think it was the experience of this particular puzzle’s play on color that brought this to my attention. Time after time, I found myself needing to open my mind to accept the possibility that pieces I never would have imagined going together actually did! 

A coping strategy for dealing with frustration when assembling puzzles is to step away from working on a difficult relationship, in order to focus on another part of the puzzle. When I am not finding the elusive connecting pieces between poinsettias and chipmunk, after searching for tens of minutes, I must move on to the yellow section of flowers on the other side. Often times, I will eventually stumble across a piece that I recognize belongs to a foreign part of the puzzle. The satisfaction felt when dropping it into its home is immense. 

Each piece has a home; Has a purpose; Has a special function. The piece that you hold in your hand belongs somewhere, even if you aren’t sure where that spot is at the moment. You can put it down, but don’t lose it! It has immense value. They all do!

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One missing piece ruins the whole puzzle! Every single one is important.

The most important dangers of a puzzler include: Do NOT lose even one piece! In fact, if you lose even ONE, you may as well throw the whole thing away. Or, if you are creative and graceful, you could make a substitute piece, but if there are more than one or two missing, it does not matter how creative or full of grace you are, the puzzle will look wholly unfinished; not holy. Never force a piece. If you do this, you could ruin its shape, so that it won’t fit nicely into the relationship it was meant for. Also, if you leave a forced piece in place, it warps the whole puzzle. You might attach a few more pieces to this wrongfully placed piece, but eventually there will be a line of unrelated-ness, like a saltwater tide infringing on brackish water in a bayou. 

IMG_2929Before I leave you to chew on this metaphor, I must mention some of its limitations. First of all, Life has no borders; It has a beginning and an end, but there is no ceiling or floor. Sore as high as you can, and be careful of exploring its depths too far. Understand that experiences, places, and people are all part of your puzzle. Also, it is more like a three-dimensional puzzle. Build out wide on either side to support potential experiences that could attach to your life. Then, watch piece after piece fall into place, grow, and assemble. Finally, but not conclusively, it is ever changing. The piece you hold in your hand may not fit now, but in a few minutes, it could be exactly what you need! Life is like a coral reef; a puzzle of plant, animal, and rock… Living rock.