Welcome to this week’s Math Munch!

I recently ran across a very ancient puzzle with a very modern solution– and a very funny name. It’s called the Stomachion, and it looks like this:

So, what do you do? The puzzle is made up of these fourteen pieces carved out of a 12 by 12 square– and the challenge is to make as many different squares as possible using all of the pieces. No one is totally sure who invented the Stomachion puzzle, but it’s definite that Archimedes, one of the most famous Ancient Greek mathematicians, had a lot of fun working on it.

Sometimes Archimedes used the Stomachion pieces to make fun shapes, like elephants and flying birds. (If you think that sounds like fun, check out this page of Stomachion critters to try making and this lesson about the Stomachion puzzle from NCTM.) But his favorite thing to do with the Stomachion pieces was to arrange them into squares!

It’s clear that you can arrange the Stomachion pieces into a square in at least one way– because that’s how they start before you cut them out. But is there another way to do it? And, if there’s a second way, is there a third? How about a fourth? Because Archimedes was wondering about how many ways there are to make a square with Stomachion pieces, some mathematicians give him credit for being an inventor of *combinatorics*, the branch of math that studies counting things.

It turns out that there are many, many ways to make squares (the picture above shows all of them– click on it for greater detail)– and Archimedes didn’t find them all. But someone else did, over 2,000 years later! He used a computer to solve the problem– something Archimedes could never have done– but mathematician Bill Cutler found that there are 536 ways to make a square with Stomachion pieces! That’s a lot! If you’ve tried to make squares with the pieces, you might be particularly surprised– it’s pretty tricky to arrange them into one unique square, let alone 536. This finding was such a big deal that it made it into the New York Times. (Though you may notice that the number reported in the article is different– that’s how many ways there are to make a square if you include all of the solutions that are symmetrically the same.)

Other mathematicians have worked on finding the number of ways to arrange the Stomachion pieces into other shapes– such as triangles and diamonds. Given that it took until 2003 for someone to find the solution for squares, there are many, many open questions about the Stomachion puzzle just waiting to be solved! Who knows– if you play with the Stomachion long enough, maybe you’ll discover something new!

Next up, the mathematicians over at Numberphile have worked out a solution to a problem that plagued me a few weeks ago while I was camping– choosing the best outdoor toilet to use without checking all of them for grossness first. Is there a way to ensure that you won’t end up using the most disgusting toilet without having to look in every single one of them? Turns out there is! Watch this video to learn how:

Finally, a little blast from the past. Almost two years ago I share with you a video of something really awesome– a computer made entirely out of dominoes! Well, this year, some students and I finally got the chance to make one of our own! It very challenging and completely exhausting, but well worth the effort. Our domino computer recently made its debut on the mathematical internet, so I thought I’d share it with all of you! Enjoy!

Bon appetit!

That was educational.

Really helpful. I’ll have fun searching for a toilet at the next fair I go to

that is really smart to use dominoes as ones and zeroes, to make a domino calculator.

Thanks! Dominoes don’t make for a very practical calculator, but I definitely know a lot more about the theory behind how computers work now that I’ve made one out of dominoes. Glad you liked it!

this video is really good it is a fun way to teach using the domino

The video showed me the use of binary code along with the use of dominos to express the outcome and the process of using it. I believe that being able to make binary code out of dominos is a very creative innovation for a regular person.

the creativity was really awesome and impressive.it was entertaining to see the dominoes fall.

Hi Nate! It is entertaining to see the dominoes fall– I totally agree. It’s a little more upsetting to see them fall if you spent hours setting them up… But definitely worth it.

the mathematical way of choosing the best toilet percentage is cool. i might use it.

Hi Alex! Good luck using the toilet algorithm!

This video has shown that probability is used throughout daily life no matter the situation. Overall this method of thinking is creative but I believe it still has its flaws.

Hi Alex! Your observation about the uses of probability is a good one. I agree that this type of thinking does have some practical draw-backs, though. What do you think are the flaws? Thanks for the comment!

536 different ways? Wow, that is impressive. I wonder what computer program Bill Cutler used and how long it took him to solve this puzzle.

Hi Cyrus! Those are great questions– and I don’t know the answer to either one. Maybe one of the links in the post will lead you there? I imagine that once Bill Cutler wrote the program, solving the problem was smooth-sailing– but writing the program probably took some time. I’m not sure whether I’m surprised that there are 536 different ways because it sounds like a lot or because it doesn’t sound like a lot… What do you think?

That was an awesome way to learn math! It is very creative and i would much rather see the technique like this than in accountants!

toilet math was a great vidieo but I still dont understand the prosess. I think it was a good idea to grab kids attention by using the topic of toilets. I think the vidieo should have been more thourough.

the domino documentary was awesome! it really impresses me that you can make a calculator out of dominos!

i liked the toilet math though i still dont quite understand the process. I liked that you grabbed the viewers attention by using toilets as a subject instead of secrataries.

Thanks, Tay! Toilets are definitely more interesting than secretaries. Finding a not yucky toilet is also a much more serious problem. 🙂

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How long did it take you to set up all the dominoes in the first clip? And did you guys mess up making the dominoes?

Great questions, Gabriel. It took us HOURS to set up all of those dominoes– at least 3. And, yes, we did mess up. Maybe you caught the calculation mistake… As you might imagine, it’s really easy to accidentally knock over a large portion of the calculator while you’re building it. We did that a few times and didn’t set the dominoes back up again quite as precisely as we should have… Despite the errors and the immense time commitment, it was still a very fun project! I highly recommend it.

I thought that the different way of adding was very interesting and I had never heard of it before. I want to learn more about it!

That’s so cool

The calculator needs to find more of a way to include somewhat of an easier general idea. Instead of doing an AND try and OR possibly?

Hello! this is really confusing i still am trying to convert some other base 10 numbers into binary numbers would 10 be 2.5.00 or what i am very confused but this is really neat did some dominos not get nocked over like they were supposed to or did one dominoe get pushed over that wasn’t supposed to?

I found this video very cool because not only did they explain what binary is they made a domino binary calculator and explained what will happen when the either push both dominos are just one .

The video about a mathematical way of choosing a toilet to use was pretty cool. I liked the idea of chance and how it played out in the numbers. Also, I think that it’s one of the most helpful things math can teach.