Godzilla is celebrating Star Wars Day with a Severed Wampa Arm Cake (recipe from justJenn):

Ewwww — that looks kind of gross. Godzilla is such a monster. Here’s a less bloody angle:

May the Fourth be with you!

12 tables, 24 chairs, and plenty of chalk

Godzilla is celebrating Star Wars Day with a Severed Wampa Arm Cake (recipe from justJenn):

Ewwww — that looks kind of gross. Godzilla is such a monster. Here’s a less bloody angle:

May the Fourth be with you!

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Godzilla is a well-known mind-reader, and in honor of final exams, which are coming up sooner than seems possible, he’d like to demonstrate his powers. Even over the internet, because his powers are MIGHTY. Like him.

Start with 3-digit number that is not a palindrome (so 360 is OK, but 363 is not). Then reverse the digits, and subtract the smaller number from the larger. You get a **NEW AND IMPROVED** number. So if you do start with 360, your **NEW AND IMPROVED** number will be 297 (which is 360 minus 063).

Treating your **NEW AND IMPROVED** number as a 3-digit number, reverse the digits. This means that if your **NEW AND IMPROVED** number appeared to only have two digits, or even one, then you have to tack on one or two leading zeros that you include in the reversal.

Now add your **NEW AND IMPROVED** number to its reverse. Godzilla will now tell you the sum, even over all the miles and electrons that separate you from this friendly beast…..

See Mini-G look at this fine piece of stripey art:

Isn’t that interesting, full of nuance? NO — it looks totally boring. But Mini-G is actually looking at it at an angle, which turns out to be a completely different story.

No more simple stripes! And while it’s no Mona Lisa*, it’s pretty cool to see the shapes appear just as you start to walk away in search of something less vertical to look at. Even better, it’s simple knitting. REALLY simple knitting, just knits and purls, where using stockinette stitch makes a color fade into the background when viewed from the side, and using garter stitch makes a color stand out. There’s a great explanation here, where “great”=“uses legos”.

This comes from Woolly Thoughts (“In pursuit of Crafty Mathematics”) and their newish illusion site. It’s a free pattern — Woo hoo! — and not that I’m suggesting that you knit during meetings or anything, but if you DID knit during meetings this particular pattern is simple enough that you can do it without being distracted from the Important Conversations and Presentations, and then you can feel good at the end of two hours that you made quite a bit of progress on your knitting whether the meeting led to a resolution or not, plus you get to point out that you’re really doing mathematics if anyone asks what you’re knitting. Win-win!

* though there is a pattern for that.

No math today (except the usual fractions in cooking): the purpose of this post is to introduce you to Godzilla’s young ward, who currently (but not necessarily permanently) goes by the name Mini-G. Little Godzilla is not very tall:

but this means he gets to travel more, and do things like ride the bumper cars at amusement parks.

Today (well, actually, last weekend) Big-G and Mini-G teamed up to prepare a delectable feast of a dessert that is dangerously easy to make: Chocolate Éclaire Pie, the recipe of which is due to our no-longer-brand-new faculty member Nicole. (Actually, I think the recipe comes from her mom, and possibly someone else before that. Thanks Nicole’s mom and anyone else it might have come from!)

Here’s how you start:

1 box of Graham crackers

2 small boxes of vanilla instant pudding

1 (8oz) carton of Cool Whip

3.5 cups of milk

Grease the bottom of a 9×13 cake pan (so maybe this is really a cake and not a pie) and line it with graham crackers. You’ll use just under 1/3 of a box, so you’ll have some left over for snacking.

Mix the pudding with milk and beat it about about 2 min.

Then blend in the cool whip. You’re practically done with the cake-part now!

(There’s about an 80% chance that someone was singing, “Whisk it! Whisk it good!” during the filming of this portion.)

Pour 1/2 the mixture over the crackers. Add another layer of crackers and cover the rest with pudding mixture, then add a final layer of crackers to the top of that.

Cover it, and refrigerate it at least two hours. When it’s about ready, it’s time to make the Topping!

¾ cup cocoa

¼ cup oil

4 tsp vanilla

3 cups 10x (powdered) sugar

6 Tbsp (3/8 cup) milk

6 Tbsp butter or margarine, melted

4 tsp white karo syrup

Mix all the topping ingredients together until smooth. [An electric mixer would have been more effective, but harder for the Gs to hold.]

Spread on top of the graham crackers, and refrigerate it for 20 more hours. And now it’s ready!

YUMMMMMMMMM!!!!

**Note 1: ** The original recipe only had half the amounts for the topping, but Nicole confessed that she always doubled it, so who am I to argue?

**Note 2: **When I made it the topping came out a little thick, so it might be OK to add a little more milk if it seems that way to you too.

**Note 3:** I’m not totally sure about that final 20 hours; I keep meaning to double check that it’s not a typo. But I kind of like the 20 hours because it means you have to make it a day in advance, which works well if you’re having people over or bringing it somewhere.

Suppose you’re on the open seas, and your GPS falls overboard. Or maybe it’s 400 years ago. How can you tell what your latitude and longitude are?

We’ll start with latitude, because it’s easier. A LOT easier. If it’s a nice night, and you’re somewhere in the Northern Hemisphere, the easiest way is to take a gander at the North Star using an astrolabe. Some astrolabes are simple, others are complicated, but they can all be used to find the North Star.

Hey, how about if Godzilla shows us how to use a homemade one!

This model was originally designed by…someone. I don’t actually know who came up with it. But all you need is a protractor, a straw, tape, string, a weight, and someone to drill the hole in the protractor so you can attach the string [or you can use a paper protractor].

In theory, the bottom of the straw should be held up to the eye. Godzilla’s short arms aren’t really conducive to holding things up (he’s had to survive using his brains as much as his brawn), so here’s a drawing of what it would look like:

When you look through the straw at an object like the top of a tree or the North Star (Hmm, why didn’t I draw a star? Because Godzilla was actually facing tree in our yard at the time, so I got distracted). Anyway, when you look through the straw, the weight causes the string to fall right at the angle that the object is above the horizontal!

Here’s why:

So sailors just grab their handy dandy astrolabe, measure the distance of the North Star above the horizon, and that’s the latitude! [I always have to think that one through. On the equator, the North Star would pretty much be on the horizon, so that’s 0° above the horizon, and at the North Pole the North Star would be directly overhead, which is 90° above the horizon.]

When Godzilla isn’t trampling buildings, flipping pancakes, or making cookies, he likes to engage in the fiber arts.

So he decided to crochet a hexaflexagon. This is a hexagon that seems flat, but can be twisted to show hidden sides.

Here’s the hexaflexagon that Godzilla made using a pattern from Woolly Thoughts (link updated 1/1/10). It initially looks like this:

but he can twist the inside…

and there is purple in the middle instead of blue sparkles!

Then he can twist it again…

And it’s orange in the middle!

And those aren’t the only colors. If you look at the other side, there is this

and this

and, finally, this!

(Crocheting it can go pretty quickly, depending on how many meetings or TV shows are on your schedule; you can also make paper versions using patterns from here or here or, of course, here.)

Last week I had two guest speakers in my *Thinking Mathematically* class. They came to talk about how math is used in the fiber arts, particularly clothing design. For example, shawls made in different places combine simple geometric shapes in different ways to come up with styles that serve different purposes, depending on the styles and needs of the people involved (only they said it without using the word “different” three times in a single sentence because they’re professionals). Or how to use proportions to design a sweater, or even how to use the relationship between the circumference of a circle and the radius to create a design for a circular shawl by doubling the number of stitches whenever the number of rows has been doubled.

At the end, they showed me a doohickey called a McMorran Yarn Balance. Both Mary Louise and Meg are spinners as well as knitters and dyers, and when you spin your own yarn, this McMorran Balance can tell you how many yards of yarn you’re getting per pound!

Godzilla is here to show you how it works.

First, you take the Yarn Balance out of the box and set it up. It’s a rectangular prism with grooves for the balance part. There’s a notch in the balance part itself, and that has to be face up because it’s where the yarn will go.

Then you take a piece of your yarn and drape it in the slot. Get out your scissors, because you’ll need those next!

Now carefully trim the yarn until the balance part is completely balanced.

Now you take that little piece of yarn that is perfectly balanced, and you measure it in Inches. Because this in an Imperial Yarn Balance (which sounds a bit like Imperial Storm Troopers, but there are none of those around. And if there were, Godzilla could totally take them.). They also sell Metric Yarn Balances.

This piece of yarn measures 4 inches. Multiply that by 100 {that’s the mathy part of this post} to get 400, and that’s how many yards of this yarn there are in one pound! Most yarns actually have a higher number of yards per pound, but you can see that this yarn is kind of thick so it’s a little heavier (meaning that it doesn’t take as much length until you have a pound).

I totally wish that I were the person who invented this balance and figured out exactly where the notch would have to be so that you only have to multiply by 100. Sadly, I can’t find anything about the history of this device online, so I don’t know who to thank for this little creation!

*Thanks to Mary Louise and Meg for coming to my class again and sharing your talents!*

Today is the last day of the Niagara International Kite Festival, celebrating all things kite (including the fact that they had to use a kite back in 1848 to start building the first bridge over Niagara Falls. Seriously — there was a Kite Contest and it took 8 days to get a kite to fly from the Canadian side to the US side; once the kite had made it across a stronger string was attached, and then a cable, and from there they had a connection and could send bridge stuff across.)

Godzilla wasn’t able to make it to the festival itself, but he did celebrate by making a kite. He’s pretty crafty so he could have made a really complicated one, but he decided to go simple with this round and use the pattern from Big Wind Kite Factory in Moloka’i, Hawai’i. [This is designed as a classroom project for younger kids, and if you pre-cut the string and use a hole-punch you can avoid scissors altogether. The original instructions have more detail, but no giant lizards.]

To start, Godzilla took an ordinary piece of paper. The original design was for an 8½”×11″ piece of paper, but he used a slightly heavier construction paper because it had subtle glitter built in. Godzilla may be a monster, but he appreciates a good glitter paper.

He sat down and folded it in half width-wise. Apparently this is called a hamburger fold (as opposed to a hot dog fold, which is when you fold it so that it becomes long and skinny, like a hot dog bun).

The fold is on the right-hand side of the paper (on Godzilla’s left). He folded a slight diagonal near that side of the paper. This formed the spine of the kite.

Then Godzilla unfolded it and admired the result.

He taped the fold together, because taping is fun.

The directions at this point say to tape an 8″ dowel across the top. But cooking skewers are a lot cheaper, and he already had those in the cupboard because Godzilla is a bit of a gourmet chef.

This would cause a problem with kids — not just the length, but the fact that it’s a pointy object that would become a weapon in about 3 seconds if given to (our) kids. If this were done with children, I’d recommend chopping the end off to size before starting the whole kite-making. Godzilla, however, had a more unique way of making sure the stick was the right length:

The next step was to attach the string (unless you want to have a paper airplane. Our 8-year old did test it at this point, and it made fancy moves but didn’t really fly very far….). You can use scissors, a hole punch, or your talons to make a hole in the spine to tie the string.

Now, in theory, you attach a tail. Godzilla skipped the tail part because he was afraid if he took any more time this blog post would never get written, and he went straight to the flying. Fortunately there was a nice stiff breeze in the living room — just look how well his kite flies!

**Edited 10/6 to add:** I meant to mention that I used this kite design in an Inquiry workshop for teachers this summer. Since you can vary it (or other kite designs) pretty easily — using different weights of paper, changing the fold, adding weights — it’s a great opportunity for kids to create their own experiments to find out which kite flies the longest, the highest, etc. That’s Godzilla: helping science students everywhere!

Here’s a slightly more impressive math card trick than yesterday. Start with a full deck of 52 cards [or, rather, start with any 52 cards — it’s not a problem if they come from a mixture of decks]. Give the deck to a Volunteer to shuffle, and afterwards have the Volunteer take the top 12 cards and give you the rest. Sometime during the next step you should sneak a peek at what the bottom card is in the rest-of-the deck. This can best be done by holding your hand casually so the bottom card is face-up enough for you to see, but it looks like you’re just holding them up before placing them down at the end of the next step.

The Volunteer should then look at their 12 cards, not showing them to you, and pick any four of them. [This is a good time to be glancing at the bottom card in the pile you hold in your hand!] Those four get laid face-up in four separate piles on the table, and the poor unchosen eight get put face down in another pile. You put the rest of the deck on top of that pile.

Now think a moment, perhaps idly shuffling the top bunch of cards in the pile [just not the bottom 9], and say that you are going to have the Volunteer lay out some cards and you will predict the last one. Concentrate, and then write the name of the bottom card that you so sneakily looked at before on a slip of paper, fold it before the Volunteer can read it, and give it to the Volunteer for safekeeping. Also give the Volunteer the remaining pile of 48 cards.

Now the Volunteer should look at each of the face-up cards, start with that number, and place cards face-down on each while counting up to 10. For example, if one of the cards is a 7 then the Volunteer would place 3 cards face-down on top, counting “8, 9, 10”. Picture cards count as 10 already, so no cards would be placed on top of those. Be sure that you can still see the original face-up cards.

When all four face-up cards have piles on them, have the Volunteer add the values of the face-up cards. For example, if the cards were a 7, a King, an ace, and a 4 then the total would be 7+10+1+4=22. The Volunteer should count off that many cards face-down from the top of the remaining deck, and then turn over the very last card (the 22^{nd} card, in the case above). Have the Volunteer then look at the folded paper that you wrote on earlier, and Lo and Behold you’ve predicted that very card! Woo hoo!

*Like yesterday’s trick, this one is a variation from card-trick.com. Thanks site!*

It’s a little known fact that Godzilla likes to throw dinner parties. Some of these gatherings are formal dinners, with no fewer than four forks, but others are more intimate. Upon occasion Godzilla throws a dinner with only two guests, but then at the last minute a friend drops by and naturally Godzilla invites him to stay for enchiladas with mole sauce (or whatever the evening’s menu). This means that Godzilla has to add a fourth place setting to his triangular table. What to do?

Fortunately, Godzilla’s table is hinged and can turn from an equilateral triangle into a square at a moment’s notice. Allow him to demonstrate. In the mock up below the separate hinged pieces are colored for easy demarcation.

Godzilla prepares by looking at the table.

He slowly starts to separate the pieces:

You can see where the three hinges are (on the outside of the triangle) connecting the four pieces. Godzilla continues to spread them out. There’s a pretty star outline in the center.

He swings those bottom three pieces around…

and up towards to the top

Now it’s starting to look like a square:

And voilà! He’s gotten a square table! The party is saved!

Here’s the final formation in color.

Isn’t that cool?

There’s a slight problem with this design, though, in that there have to be at least four table legs, all close together. Fortunately Greg N. Frederickson designed a triangular table with a large enough piece in the center so that a single pedestal would do, and the six tiny swinging pieces could all be hidden with a linen table cloth. It’s the lead article (“Designing a Table Both Swinging and Stable”) of this month’s *College Mathematics Journal*. He has some nifty spiff animations here and even more information at the bottom of this page.

For more on these dissections, check out Ivars Peterson’s January 27, 2003 Math Trek column about how chemists were doing exactly the same sort of table spinning as Godzilla (without mentioning him by name, of course) using little plates that would self assemble into different kinds of shapes (although to do the triangle-square hinged switcharoo they had to be connected with thread.)

Time for the dinner party. Pass the enchiladas.

The answer, naturally, is 15. If you’re talking about the Burnt Pancake problem, that is. (And the sequence actually starts 1, 4, 6, 8… but I left off the initial 1 because otherwise you would have known right away that something was amiss.)

The Burnt Pancake problem involves pancakes of different sizes, each with one burnt side, piled up on top of one another. Here’s how famous math guy and Emmy winner David X. Cohen initially described the problem in this interview with Sarah Greenwald:

The question was how do you sort these disks to get the biggest pancake on the bottom and the smallest pancake on top [with all the burnt sides down] if they start in an arbitrary disordered state, and the only thing you’re allowed to do is put a spatula somewhere in the middle, pick up the ones above it, flip them over, and put them down, as a group. Doing that repeatedly, putting a spatula in different places, you want to sort this out. So a very physical thing, that got me excited when I found out that no one knew the answer in general for how many flips it takes to sort this thing.

Here is help us visualize this problem is our friendly neighborhood Godzilla. He’s going to use oreos with tops removed to simulate the pancakes.

With only one “pancake”, if it starts like this

the “burnt” side is already at the bottom, so it needs 0 flips to get into the proper position.

If, instead, the pancake starts “burnt” side up, it needs one flip:

(Don’t you think that Godzilla looks a little bit like that guy Craig from *Hell’s Kitchen*?) As the Big G has just demonstrated, if you have just ONE burnt pancake then it could take as many as, well, 1 flip to orient it correctly. That’s where the 1 in the sequence 1, 4, 6, 8, … comes from.

Now let’s look at what happens with two burnt pancakes. We want to end up with the larger pancake on the bottom, and all the burnt sides down like this:

But what will happen if the pancakes start off in a different configuration? How many flips have to be done? It turns out that it could be as many as 4 flips:

Suppose your pancakes start off in this pile: , with the burnt parts on top. The first thing Godzilla does is to put the spatula on the bottom and flip the pile over.

Now the pancakes have the burnt side down, but the smaller one is on the bottom. The big top pancake will have to be flipped:

(Godzilla’s being a little sloppy here: he should only be picking up the top pancake.) After this maneuver the burnt parts are on the “outside” but big pancake is still on top. The entire stack needs to be flipped to get the little pancake back on top:

Now the little pancake is on the top and the burnt parts are on the “outside”, so the top pancake must be flipped:

And now voilà, the pancakes are in the right order!

See how happy Godzilla is! He knows he gets to eat these oreos when all is said and done.

With a different starting configuration (there are 2!·2^{2}=8 ways they initially could be piled up, with the smaller pancake on the top or the bottom, and the various burnt sides up or down), it turns out that it will take at most 4 flips to get them in the correct order. That’s where the 4 comes from in 1, 4, 6, 8, ….

What happens if you start with three pancakes of different sizes? The desired ending configuration is this:

There are 3!·2^{3}=48 different ways the pancakes could start out. Some of them could be turned into the proper configuration after just one flip:

but other configurations require more. This one, for example:

The cookies are in the right order, but it just takes a lot of maneuvering to get the burnt parts on the bottom instead of the top: 6 flips. That’s the most it would take no matter how the pancakes started out.

So for one pancake it could take as many as 1 flip, for two pancakes it takes up to 4 flips, for three pancakes up to 6 flips, for four pancakes up to 8 flips, for five pancakes up to 10 flips, for six pancakes up to 12 flips, and for seven pancakes up to 14 flips.

Then for eight pancakes, it only takes up to 15 flips. And for nine pancakes 17 flips, then for ten pancakes it goes up to 18 flips (according to the Online Encylopedia of Integer Sequences). But by 11 pancakes there are over 81 billion different initial configurations, so checking by hand to find the smallest number of flips for each configuration is tough. Fortunately, as described in this earlier post, we have E. Coli to help us figure it out. [They give the E. Coli a configuration, let them do a specific number of flips, and any E. Coli that get the virtual pancakes in the right order become resistant to the antibiotic tetracycline in honor of their good effort. Then the scientists add some tetracycline and if any of the E. Coli survive, they know that the pancakes could be put in order after that particular number of flips. The animation E. Hop gives the whole scoop.]

Meanwhile, Godzilla is going to take a little break. Bon appetit!