# God’s Number, Chocolate, and Devil’s Number

Welcome to this week’s Math Munch! This week, I’m sharing with you some math things that make me go, “What?!” Maybe you’ll find them surprising, too.

The first time I heard about this I didn’t believe it. If you’ve never heard it, you probably won’t believe it either.

Ever tried to solve one of these? I’ve only solved a Rubik’s cube once or twice, always with lots of help – but every time I’ve worked on one, it’s taken FOREVER to make any progress. Lots of time, lots of moves…. There are 43,252,003,274,489,856,000 (yes, that’s 43 quintillion) different configurations of a Rubik’s cube, so solving a cube from any one of these states must take a ridiculous number of moves. Right?

Nope. In 2010, some mathematicians and computer scientists proved that every single Rubik’s cube – no matter how it’s mixed up – can be solved in at most 20 moves. Because only an all-knowing being could figure out how to solve any Rubik’s cube in 20 moves or less, the mathematicians called this number God’s Number.

Once you get over the disbelief that any of the 43 quintillion cube configurations can be solved in less than 20 moves, you may start to wonder how someone proved that. Maybe the mathematicians found a really clever way that didn’t involve solving every cube?

Not really – they just used a REALLY POWERFUL computer. Check out this great video from Numberphile about God’s number to learn more:

Here’s a chart that shows how many Rubik’s cube configurations need different numbers of moves to solve. I think it’s surprising that so few required all 20 moves. Even though every cube can be solved in 20 or less moves, this is very hard to do. I think it’s interesting how in the video, one of the people interviewed points out that solving a cube in very few moves is probably much more impressive than solving a cube in very little time. Just think – it takes so much thought to figure out how to solve a Rubik’s cube at all. If you also tried to solve it efficiently… that would really be a puzzle.

Next, check out this cool video. Its appealing title is, “How to create chocolate out of nothing.”

This type of puzzle, where area seems to magically appear or disappear when it shouldn’t, is called a geometric vanish. We’ve been talking about these a lot at school, and one of the things we’re wondering is whether you can do what the guy in the video did again, to make a second magical square of chocolate. What do you think?

Finally, I’ve always found infinity baffling. It’s so hard to think about. Here’s a particularly baffling question: which is bigger, infinity or infinity plus one? Is there something bigger than infinity?

I found this great story that helps me think about different sizes of infinity. It’s based on similar story by mathematician Raymond Smullyan. In the story, you are trapped by the devil until you guess the devil’s number. The story tells you how to guarantee that you’ll guess the devil’s number depending on what sets of numbers the devil chooses from.

Surprisingly, you’ll be able to guess the devil’s number even if he picks from a set of numbers with an infinite number of numbers in it! You’ll guess his number if he picked from the counting numbers larger than zero, positive or negative counting numbers, or all fractions and counting numbers. You’d think that there would be too many fractions for you to guess the devil’s number if he included those in his set. There are infinitely many counting numbers – but aren’t there even more fractions? The story tells you about a great way to organize your guessing that works even with fractions. (And shows that the set of numbers with fractions AND counting numbers is the same size as the set of numbers with just counting numbers… Whoa.)

Is there something mathematical that makes you go, “What?!” How about, “HUH?!” If so, send us an email or leave us a note in the comments. We’d love to hear about it!

Bon appetit!

### 90 responses »

1. That was pretty hard to comprehend. I understood most of it. The amazing part about this video was the superflip. The least moves you can do the superflip in is 20 moves. I thought that was cool.

• I really want to try the geometric vanish the next time I get a chocolate bar. I also liked the rubix cube video because I am someone who is not very good at solving rubix cubes so I was amazed and the fact that people can solve the puzzle with only 20 moves .wow! I am definitely going to try to try to solve the rubix cube using math. I have never thought about using math for it.

• It’s interesting how you say that you’d never thought of using math to solve a Rubik’s cube. I’d never really thought of it, either. I always thought you just had to memorize a bunch of different steps – which I was never very good at doing. But solving one in only 20 moves would require some very different (and mathy) thinking. If you try, good luck!

2. As someone who is able to solve a Rubik’s Cube, not very fast but I can do it, I was interested in seeing if there was a way I could do it faster. This video was cool because I had heard of the God Number before but had never really looked into it and this video shed a lot of light on the subject. I find it amazing that any Rubik’s Cube can be solved in 20 moves, I hope one day we will be able to find these moves and the whole world will be able to do the Rubik’s Cube.

• Wouldn’t it be awesome if we knew how to solve every Rubik’s cube in 20 moves? How hard would it be to do that, though??? There are SO MANY different configurations… We can dream, though.

3. The video How To Create Chocolate Out of Nothing really confused me. I have watched it numerous times and I cannot find how it just “vanishes”. I believe that if you were to repeat the process, it would manifest another piece of chocolate because it is the same figure as the original, so theoretically, would it work infinitely? I guess eventually there won’t be enough chocolate to make the process work. Very interesting though.

• It’s super tricky! I suggest trying it yourself using graph paper. That’s how I eventually fully understood what was going on. (You could try it with chocolate, but you’d probably make a mess… could be fun, though.)

4. I think the chocolate trick is amazing and I cant figure out how it works

5. I tried this experiment with paper and every time I took a piece of paper off, the paper got shorter by a quarter of the blocks height, but in the width of the paper.

• Exactly – that’s what my class of 6th graders found as well. The video is sneakier because the row of chocolate squares that shrinks is actually a row of rectangles to start with – so it’s a bit more obvious something is amiss with graph paper. We then wondered if the trick would work better on rectangles of different dimensions… That’s worth experimenting with!

• Hi Pat! I’ve actually seen your blog post – I looked at it while I was writing this one. :) My 6th graders and I are about to study the Hooper/Guyot puzzle. Thanks for your great research and writing!

6. in the beginning it was 6 by 4 and at the end it stayed that same. it was really interesting how 1 piece of chocolate was left i watched the video a lot of times, until i could kinda see what was happening.

7. Wow, this is really an amazing post. I had never understood the term God’s number until I saw this. The video describes very well about God’s number and how they found God’s number to be twenty. I also was shocked by the chocolate video. How did he do that? I’m interested to learn more about this topic and if he made a chocolate piece appear, or disappear. These videos and posts are absolutely amazing, I really learned something today.

• Hi, Karisa! That’s awesome that you learned something! I didn’t really understand how they found God’s number either, and I think it’s crazy that they basically just used a mega computer to solve every Rubik’s cube. And for the chocolate puzzle, I suggest you try it yourself. :) Thanks for commenting!

8. Both those videos were very interesting especially the “How to create chocolate out of nothing” video. My first time seeing the video I was curious how he managed to remove a rectangle and keep a rectangular shape. The second time I watched the video I understood how he slid the shorter piece of chocolate to a higher position and vise versa with the bigger piece of chocolate.

I never really heard of the term “God’s Number” before, but I understand it now. It was fascinating how many possible moves you can do with a Rubik’s Cube. I would have never thought of a number that big!

9. I feel like the spaces that were left in between the chocolate pieces were actually constituting for the missing piece of chocolate and in reality you weren’t actually getting extra chocolate it is just an illusion.

10. For the chocolate post, if you look clearly, then you can see that a bit of chocolate is cut off when you switch the pieces. Therefore, the chocolate will eventually disappear.

11. The Rubik’s cube video was very interesting. I have seen friends bring Rubik’s cubes to school and show me how fast they could solve them. I was really surprised to find out that they were using an algorithm to solve them so quickly. I thought that the superflip was really cool and have also seen my friends do a different type of pattern where on all the faces of the cube they had the center square be different than the surrounding squares and it looked almost like a flower. I was wondering what the fastest record was and how many moves it took. This video made me want a Rubik’s cube to try and see how fast I could solve it. Thank you for posting this and keep up the good work.

• Hi Felix! I had a similar experience when I learned that my friends were using an algorithm to solve Rubik’s cubes. I always thought they were figuring it out as they went along… Thanks for your comment!

12. The Rubik’s Cube video was amazing I never knew math would be in it but I have a feeling that Rubik’s Cube can be solved in 15 or 10 moves but I can be wrong. How did they come up with the name God’s Number? As for the chocolate video I never knew that one little piece could make a huge difference. These two videos were great!

• Hi Cristian! I think it’s called God’s Number because only an all-knowing, super smart being (like a god) could solve every Rubik’s cube in at least 20 moves. Thanks for commenting!

• I sall the chochlet video it was cool how he took the little piece out and he made the same shape

13. The chocolate post was very cool it was like he was saving chocolate I really liked it!!!

• Wouldn’t it be cool if you could save chocolate for later and eat all your chocolate at the same time???

14. I know some things about Rubik’s Cubes, because I know how to solve one. The amount of possible permutations for a Rubik’s Cube are gigantic. I wonder where they started, the people reducing the scenarios needed to be checked. It seems so overwhelming I wouldn’t know where to start.

• Hi Ryan! That’s cool that you know how to solve a Rubik’s cube. Starting to solve ALL THE RUBIK’S CUBES does sound very overwhelming. I think they started by trying to make the number of cases they needed to solve smaller – by finding similarities between Rubik’s cube permutations so that they could solve just one permutation and apply the solution to many more. That would make the process less overwhelming… but still enormous. Thanks for your comment!

15. WOW that is so cool i am going to do that some time ! this video really makes me want chocolate and ill just tell my parents i cut the chocolate. i hope this could work again and again and again

16. Hi, I have two qustions for you guys the first one is,”Can you use more than two piceses of chocolate and put them into one or would you have to use one picese of chocolate?” my second question is, “Do you move one pieces every time you cut a piece?”

17. That video was really complicated but since me and my brother watched it( he knows how to solve the rubiks cube) he explained the rules of the cube, and the guy who was talking really explained it well though it still didnt run through my brain correctly cause im not the kind of person that gets really complicated stuff.We both enjoyed the video and im gonna ATTEMPT to learn it.I repeat i will ATTEMPT to. It really cool to see that the rubiks cube was doing some work for charity. Very interesting video and ill try to learn the cube. Thank you and keep it coming. :)

18. I really enjoyed watching this since I know how to solve the rubiks cube. I never knew that it is possible to solve the rubiks cube in 20 moves, that is just unbelievable. I like that they are doing work for charity with the rubiks cube. I will try to memorize the super flip and try to solve the superflip as well. It is very interesting how there it can be solved in 20 moves and at other times it can be as big as 29 quintillion different moves.

19. Why in the video they say gods number is 20 but say you can solve a rubix cube in or fewer? Do they mean like if you get a new rubix cube and just make one detail then solve in one?

20. The video that I watched about the geometric vanish was very interesting. I’m going to try it the next time I get a chocolate bar! One question that I have is can you do what the guy in the video did again and still keep that many pieces of chocolate?

21. I find this video very interesting, I myself have challenged the rubiks cube (I may have nearly completely failed) and my estimation for how many turns it took to solve was at least 75-100 in my own trials, but don’t quote me on that. What got me into the rubiks cube was its difficulty and its mechanics, but after watching this it amazed me at how no matter what the problem is, it can be solved with significant thinking power. I am curios as to what the program is that was ran on googles computer system and who coded it. What else can we learn about the rubiks cube? And how can we use what we learned from this around the world?

22. I found the “How to Create Chocolate out of Nothing” video very interesting. It was done on a 4×6 grid of chocolate, and I was wondering if it would still work on a different size grid, and if it did, how you would adjust the 2-3 cut as well.

23. When i watched the “Making Chocolate Out of Nothing” video I was amazed at how the chocolate simply just “vanishes” and i have looked at this video many times, yet I can’t figure out how it works. I will try this soon with some chocolate, or just some paper if I can’t get any chocolate to do this with. I was very intrigued by how it just “vanished”.

24. I watched the “Making Chocolate Out of Nothing” video and was very amazed at how taking one piece of chocolate out and mixing it up like he did in the video can still make a rectangle. Me and my bro watched it like 3 times over again< i didnt get it but he did (hes in 8th grade geometry) and tried to explain it to me but I still didnt get it and I kept on trying it on a piece of paper and it got a little easier but it was still a little hard to comprehend. My appetite is not stuffed yet so ill keep on munching. :)

• Hey Daniel! It’s funny – even when you make it happen, it’s still hard to understand! Good luck working through it and keep munching! (On chocolate, too.)

25. The video “How Create Chocolate Out of Nothing” was very confusing for me. I can not get how the chocolate just vanishes no matter how many times I look at it. Next time when i get a chocolate I will try the trick and see if it actually works. Like Amy, I am wondering if the grid has to be 4×6.

26. In the “God’s Number” video I don’t understand how they can us symmetry to reduce the number of configurations because both 9 and 27 are odd numbers. What do they mean by using symmetry to reduce the number of configurations? How do they use a computer to solve Rubik’s cubes?

• Hi Baylor! I think they mean that some Rubik’s cube configurations that may look different are actually solvable in similar ways. For example, say you have a mixed-up Rubik’s cube and you’re holding it with a particular face forward. If you spin it so that a different face is forward, the steps to solve it will be the same. Showing that two Rubik’s cubes can be solved in the same way by spinning one so that it looks just like another is an example of using symmetry to reduce the number of configurations. Does that help? To try to answer your second question, I think they write computer programs that try doing different things to Rubik’s cubes until they’re solved. I don’t know much about computer programming, so I can’t give you a better answer than that – but that’s the gist of how it’s done. Thanks for your questions!

27. In “How to Create Chocolate Out of Nothing” the thing that immediately caught my eye was the angle of the first break. What is the angle of the first break? If you were to use this angle on any sized rectangle of chocolate would it work the same way?

• That’s a great question! Go ahead and try it… :)

28. I have solved a Rubiks cube about 30 times over last summer and spent so much time memorizing algorithm after algorithm and i felt pretty confident with solving the Rubiks cube but i feel like it would be very interesting how to solve the cube in only 20 moves. If you repeatedly do those 20 moves on a rubiks cube will it keep scrambling and resolving?

• Hi Dru! That’s a great question. Would you do the 20 moves in the same order after you solved the cube? I bet you could do them backwards and get the same scramble you started with.

29. I liked the way how they made the choclate pieces equal with taking a pice out. if i did it i would do it with dark choclate and instead of taking a pice out i would eat a piece. I think today after school i am going to do it.

• Mmmm dark chocolate… Lots of my students complained about the white chocolate, too. Great idea!

30. I thought the “How to Create Chocolate Out of Nothing” was very creative. “Geometric Vanish” was the confusing part. You originally start with a 6 by 4 and end with a 6 by 4 with one Nothing square?? Maybe I need to watch it again. If Geometric Vanish works with a 6 by 4 will it work for any other number so you can have 2 squares left over?

31. That was really amazing what you did with choclate bar i acctually had a chocolate handy and did it for myself. Soon after I realized how it was done, but it was still really amazing. i think I remember seeing this video on an app I have. The questions I have are would this work if you hade ant size or shape choclate bar? And did you, Justin, or Paul make this video?

• Hi Anthony! Your question about whether this would work on any size chocolate bar is a great one. Maybe the trick would even be sneakier if the chocolate bar were a different size… We didn’t make this video ourselves. I found it while reading another blog, actually. Thanks for you comments!

32. The “Make Chocolate Out of Nothing” was just so mindblowing. I can’t figure out how the chocolate keeps becoming a whole again! I really want to try this on graph paper and with real chocolate. I’m thinking, however, that doesn’t the chocolate have to run out eventually? Or is it infinite chocolate? This video really made me think.

• Hi Alisha! Infinite chocolate would be wonderful… but, yes, your intuition is right – the chocolate does have to run out eventually! You should definitely try it with chocolate or graph paper. You’ll see the trick it you do!

33. Its amazing that a rubix cube can be solved in 20 moves or less. Its great to see how far math has come.

34. After watching the “Making Chocolate out of Nothing” video several times, I still cannot see how it works or tricks you into believing that it works. I’ve seen this same problem before but still haven’t figured out exactly how it functions because I never really had enough time to explore the depths of this phenomenon. I plan on making a paper model of the chocolate and repeating the cutting and moving processes until it becomes obvious that the rectangle is losing some area, because I know that I won’t be able to sleep at night knowing that I don’t understand this. Overall this is a very intriguing and interesting video and I enjoyed watching it.

35. I loved the chocolate cutting trick, and I had to watch the video about 20 times before I figured it out. In the end you will notice that the chocolate is no longer touching each other at the seam. I even got graph paper out and home and found that the length shortens. I can’t wait to learn more on this topic of geometric vanishing!

36. The video Making Chocolate out of Nothing was interesting. I thought it was clever how you can cut the chocolate at certain angles and remove one cube while still having a perfect rectangle. It is hard to believe how it works.

37. The video Making Chocolate out of Nothing is very confusing, but interesting to watch. The chocolate must run out at some point so how many time can you make a square of chocolate “vanish” before this procedure doesn’t work anymore, and if the chocolate bar was larger could you make a cut 4 by 6 instead and take out two pieces of chocolate instead of one?

38. The video Making Chocolate out of Nothing was confusing, but interesting to watch. The chocolate will run out so I was wondering how many times you could do this process before it becomes impossible, and if the chocolate bar was longer could you cut is 4 by 6 instead and remove 2 chocolate squares?

39. The rubics cube is one of the coolest mathematical things in my opinion. Its just amazing what people can finish it in 20 moves. I can not do that.

40. I loved the video about how to make chocolate out of nothing although it was the second time i’ve seen it. the first time my dad showed me. anyways I loved it and plan to do it at home.

41. In the videio making chocolate out of nothing i am wondering how you can take a piece of chocolate out of the rectangle of chocolate while keeping the pice of chocolate a rectangle relatively the same size, but it is smaller than the original pice because if you take part of the whole the pice chocolate will be smaller because the result is the piece of chocolate minus the piece that was taken off.

42. AHA i have found the secret for the chocolate!! if you look closely the top pieces of chocolate are squares and the middle or obviously wider, so its a rectangle and it cuts a little bit of the area of the whole thing, resulting in the extra chocolate piece

• Haha, you got it! Little pieces of chocolate can be removed and YOU’LL NEVER KNOW because, sneakily, they were rectangles. Nicely done!

43. I really love this video and I plan on doing this at home. I’m really greatful for the comments because of them i know have a better understanding of how this works.

44. I thought it was cool how a rubix cube could be turned 43 qutillian times then be solved in 20 moves. But if you can do it in 20 moves cant you make it the same by reversing those 20 moves. I still found that fascinating that gods number is 20 because every rubix cube can be solved in 20 or less moves.

• Yeah I also found it really fascinating that the rubix cube could be solved in only 20 moves! The thing I thought was most amazing though was that some people could solve the cube just writing the steps on paper, without even touching the cube.

• I also found the chocolate video really cool! It was neat to see how every time you take a single piece of chocolate out of the chocolate bar you get the same shape just a little bit smaller. So you’re just reconfiguring the shape.

45. it was really cool how you could only solve the rubiks cube in 20 moves i also thought it was so awsome how some body could just solve it on a piece of paper i could never do that.

46. Wow. This is amazing. I LOVE how people can figure out how to solve the cube with computers. I am interested to know how that happened.

47. I watched the chocolate video at least three times it started off as 4 by 6 and ended as 4 by 6. I really liked this video it made me think! I still dont really understand it though.

48. The making chocolate out of nothing trick is really cool, but I don’t understand how it works. Logically it doesn’t make sense because each time you are taking away chocolate from the bar, but there is enough to replace it? I am stumped.

49. I thought the video was so cool because whe ever I tride to do it would take me like me like tree days to do it but it turns out you can do it in less than 24 moves.

50. I really liked the “gods number video”, I am a person who can solve the cube but it takes me a while. 20 does seem like an impossible feat to surpass. I have always been interested in the cube, and have done research on it before, and this video just informed me more about the subject.

51. I thought that the chocolate trick was really cool. I then had the thought of, “how is it possible to have infinite chocolate out of only one chocolate bar?” I tried doing the trick by myself with a Hershey’s chocolate bar. Unfortunately, when I made all of the cuts, my chocolate bar wasn’t the same size as it was the first time. as i continued to cut, it got smaller and smaller. Therefore, there is no such thing as “infinite” chocolate. I did, of course, had a lot more chocolate than the regular chocolate bar. How can chocolate just be made thin air like that? Anyways, i thought this was a great experiment and was a really fun.

52. I really liked the chocolate video and i know what a geometric vanish is, but i dont understand how it works. could you make infinite chocolate from this? Or does rectangles always have an “extra square?”

53. Awesome! i saw the video about the chocolate. I am a huge fan of chocolate! its a little confusing but at the same time its understandable! weird huh?! I cant wait till my next chocolate bar so i can try it! i’ve tried it with paper and it worked. it would be nice to share my chocolate and still have about the same amount left.

54. In the video “Gods Number” i found it was weird how it takes 20 moves to solve the most complicated mix up. I thought it would be way more because it takes me like 300 moves just to solve the easy mix up. people that can solve a Rubik’s cube in so little time are crazy cool and crazy smart!!!!!

55. I watched the chocolate out of nothing and I was wondering if you can only do that with a rectangular shape and a square shape?

56. I watched the chocolate out of nothing and I was wondering if you could only do that with a rectangular shape and a square shape

• Hi Kaitlyn! That’s a great question. I’ve seen similar sorts of things with other shapes, but nothing quite like this. Want to try? I’d love to hear what you come up with!

57. In the video “God Number” I found it interesting how one person can look at the cube and can solve it.

58. The chocolate video looks like nothing is being removed, but actually small amounts are removed each time and if you keep try it at home and keep doing it you’ll end up with no chocolate, a messy surface, and a different outlook on “magic”. Still, makes a good party trick. ^_^

59. “God Number”
-I learned that if you do 17 moves on a Rubik’s cube, the number of combinations you could reach is less than the super large number “43 quintillion”. I also learned that 20 is called “God’s Number” because only a super being could solve the rubik’s cube in 20 moves. This video interested me a lot because it would’ve probably taken me about 100 moves or more to solve one rubik’s cube, and knowing that it could be solved in 20 moves is really shocking to me!!!

60. I never really understood this trick but this video helps me understand that the extra chocolate piece came from the fourth row of the chocolate bar because the length of the bar decreases in size by a little.

61. it was cool how you could solve the rubiks cube in 20 moves i never knew that. I also thought it was super awesome how some people are extremely gifted and can look at a rubiks cube and are able to list the moves needed to solve it. i wonder how many people are actually able to do that..