A while ago I was introduced to Skyscraper Puzzles (I believe they were invented by BrainBashers). I’ll explain below about the specifics of how to play, but basically they are a great way to help our students think about perspective while thinking strategically through each puzzle. Plus, since they require us to consider a variety of vantage points of a small city block, the puzzles can be used to help our students develop their Spatial Reasoning!

I’ve written before about how to help your students persevere more in math class and I still think that one of the best ways to do this involves physically and visually thinking about tasks that involve Spatial Reasoning.

While I loved the idea of doing these puzzles the first time I saw them, I was less enthusiastic about having these puzzles as a paper-and-pencil or computer generated activity because it is difficult to help develop perspective without actually building the skyscrapers. So, I created several templates that can easily be printed, where standard link-cubes can be placed on the grid structures.

Below are the instructions for playing and templates you are welcome to use. Enjoy!

#### How to play a 4 by 4 Skyscraper Puzzle:

- Build towers in each of the squares provided sized 1 through 4 tall
- Each row has skyscrapers of different heights (1 through 4), no duplicate sizes
- Each column has skyscrapers of different heights (1 through 4), no duplicate sizes
- The rules on the outside (in grey) tell you how many skyscrapers you can see from that direction
- Taller skyscrapers block your view of shorter ones

Below is an overhead shot of a completed 4 by 4 city block. To help illustrate the different sizes, I’ve coloured each size of skyscraper a different colour. Notice that each row has exactly 1 of each size, and that each column has one of each size as well.

Below is the front view. You might notice that many of the skyscrapers are not visible from this vantage point. For instance, the left column has only 3 skyscrapers visible. We can see two in the second column, one in the third column, and two in the far right column.

Below is the view of the same city block if we looked at it from the left side. From left to right we can see 4, 1, 2, 2 skyscrapers.

Below is the view from the back of the block. From left to right you can see 1, 2, 3, 2 skyscrapers.

Below is the view from the right side of the block. From here we can see 2, 2, 4, 1 skyscrapers (taken from left to right).

When playing a beginner board you will be given the information around the outside of your city block. Each number represents the number of skyscrapers you could see if you were to look from that vantage point. For example, the one on the front view (at the bottom) would indicate that you could only see 1 skyscraper and so on… The white squares in the middle of the block have been sized so you can actually make the skyscrapers with standard link cubes.

#### Templates for you to Download:

#### A few thoughts about how you might use these:

- How will you introduce these puzzles to your students? How much information about strategies and tips will you provide? Will this allow for productive struggle, or will you attempt to remove as much of the struggle as possible?
- Would you use these as an activity you give all students, or something you provide to just some. Why?
- How would giving a puzzle to a pair of students be different than if you gave it to individuals? Which were you assuming to do here? What if you tried the other option?
- How might using physical blocks be different than paper-and-pencil or electronic versions?
- How will you orchestrate a conversation for your class to help consolidate the learning here?
- What will you do if students give up quickly? What questions / prompts will you provide?

A belief I have: Teaching mathematics is much more than providing neat things for our students, it involves countless decisions on our part about how to effectively make the best use of the problem / activity. Hopefully, this post has helped you consider your own decision making processes!

I’d love to hear how you and/or your students do!

This is the same thing that used to be on T!-73 calculators, but with a hands on approach. Glad to see it in a more concrete form.

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My students loved using this! I teach 5th grade and most of them were very engaged. I gave minimal instructions. I turned your pictures into a simple power point and then let the kids work it through. After a few solved the beginner 4×4, I had them share their strategies. One of my squirreliest students had so much fun with these. He made all the sizes he would need first then started placing them on the board. Awesome!

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Tanks for sharing Pam. I’ve heard several stories about how these puzzles have helped reach some of the hardest to reach, or the ones who struggle the most…

Nice to provide things that can help with confidence and help build spatial thinking at the same time!

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Hi Pam, would you mind sharing your Power Point for this?

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I can’t download the grids? Anyone have any ideas or another way to get them.

Thanks!

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Hi Laura,

The links were placed in the middle of the post and looked like this:

Beginner 4 by 4 Puzzles

Advanced 4 by 4 Puzzles

Beginner 5 by 5 Puzzles

Advanced 5 by 5 Puzzles

They should lead you to the PDF versions I have made for you. If you still can’t access them, send me your email address and I will send you links.

Mark

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Hello, Mark!

My students love the skyscraper puzzles and even want to spend their lunch completing them. I have a couple of questions regarding the above templates. When doing the Brainbasher version there seems to be only one solution, but my students have seemed to find multiple solutions to many of the above templates. Is this correct? We also found a 5 x 5 that does not have a solution because of the placement of a number one on a horizontal row and a four in the same column. I just want to make certain we are completing them correctly?? Please advise. I appreciate your work and response. Thank you so very much!

T.W

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Yes. There is one error I need to correct. As for multiple possible answers, I encourage you to make sure that there are only 1 of each size skyscraper in each row and in each column. Regularly I see students with finished samples where the numbers show the correct number of skyscrapers from each angle, but the city does not follow the rules of 1 size per row/column. Let me know if you still see multiple possibilities.

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I love these! Just did the 4×4 with my kids.

Wanted to do the 5×5 easy, but the link isn’t highlighted and you can’t get to it.

Could you email it to me? ngolden@merrick.k12.ny.us

Thanks!

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Never mind! Found an active link on another page of your site!

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I can’t seem to find the active link on another page. Could someone please point me in the right direction? Thanks in advance!

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Hi Stephanie, I have updated the links. They can be found toward the bottom of this article.

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Thank you! 🙂

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I’ve occasionally used these before but it’s great to have a decent set of problems to solve. It’s hard finding ways to develop spatial reasoning so this is really useful. Thanks!

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how do you play the 5 x5, buildings with 1 to 5 blocks?

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To play the 5 by 5 games. Same basic set rules apply. However, instead of there being for towers in each row and column there are now five towers in each. Yes, you are correct in that the towers are now sized 1234 and five for each row / column. If you would like, I could send you a picture of a finished puzzle. Just let me know where I can share the picture (your email or twitter account…).

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My students love these and we are stumped with the 5×5 puzzles. Can you also send me the finished puzzles? Thank you so much!

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Mark

I played with this game during TMC17 this summer. It was a great exercise for four adults to play with. I needed this reminder, I will find time soon to work this in as a class activity with my Geometry kiddos.

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I love these. Brainbashers.com has a different set every day. The first time I was introduced to them, there were no numbers outside the grid and the only instruction was to have one of each in every row and column. I think doing this helped me to focus on that aspect of the puzzle, without having to think too much about the other constraints. Plus, it was interesting to compare solutions with others in the group – there were several different versions AND each version had all kinds of patterns and symmetry!

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Making a PPT is such a great idea! I made one in Google Slides for my class (6th grade). Happy to share. Thank you Mark! https://docs.google.com/presentation/d/1rNNqEVUlHARcmpMd2sN2KnNxsV4qeevvAxgAEVgrTX8/edit?usp=sharing

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Thanks for sharing your slides. I look forward to trying this activity with my class.

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Thank you!!

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What is the youngest grade you would consider using these?

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It’s always important to know your purpose. I originally used these for grade students who were studying perspective. However, others have used these in a variety of years for a variety of purposes.

What purpose might you want these to serve?

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I am thinking of using these with 2nd graders as a problem solving task to build spatial reasoning skills.

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First I think it’s important help students imagine what something might look like from different perspectives. The book Taking Shape by Tara Flynn et., al might be helpful.

Once your kids are able to think about perspectives, these should be great.

Let me know how it goes.

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Jen, I offered them to my third grade students this year. The 4×4 were challenging but doable. The 5×5 were too hard. They often chose to do them in groups of two. One building and one checking/making suggestions.

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Am I missing something with the advanced 4 x 4? It seemed so much easier! Or is it once the code is broken it makes it easier? Thanks! Loved these!

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Keep in mind about the rules…. you are only allowed to have 1 of each size of skyscraper in any row or column. Since you have less information, it can get trickier.

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We followed those rules but the students got it so quickly I was thinking I missed something. Just did these with 5th grade and it was fantastic watching them work through the beginner puzzles. They enjoyed it just as much.

It was a lesson on perseverance 😉 Thank you so much for creating them!

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My 4th and 5th gifted students are loving these puzzles and I cannot get them to stop when the class is over! Several students want to start making their own. Thank you!

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I was wondering about the difficulty of a template that has numbers on only 2 sides vs a template that has numbers on all for sides. There is some discussion that the templates with numbers on only 2 sides is harder. I would appreciate any feedback on your findings.

Also, are there any 3 X 3 templates?

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Hi Vivienne. First, I thank you for the suggestion. I might make a few 3×3 samples soon.

As for the difficulties of each puzzle, I would recommend you try to complete a few from each set. If you disagree with the difficulties, please let me know. The idea generally is, the less information I give you, the more difficult it might be to solve.

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ok

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It’s fun to see these again. Puzzles like these were part of some research done in the ’80s to show that spatial thinking could be “taught.” See: Ben-Chaim, D., Lappan, G., & Houang, R. T. (1988). The effect of instruction on spatial visualization skills of middle school boys and girls. American Educational Research Journal, 25(1), 51-71.

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Thanks for the citation. I’ll definitely check it out.

Currently, I’ve been using Bruce, Flynn, Moss et al for their ideas for experiences to teach spatial reasoning/spatial visualization. They draw on some exciting new research.

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Please don’t misattribute the inventor of the Skyscraper puzzle. This puzzle has been around before brainbusters

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If you know the creator, I would be happy to amend my post. Currently I have been given information that Brainbashers was the originator.

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