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Numbers:
• Total balloons inflated, filling foyer & mailroom: 2500
• Balloons filling foyer only: 1670 +/- 20 (some popped)
• Our initial estimate for foyer only: 1400 (based filling a 3-meter cube with 30-cm balloons at 72% packing efficiency)
• Time to fill foyer: 3 1/2 hours with 6-8 people working at any given time

1. Make a fist around two fingers (borrowed from someone else’s hand) and squeeze hard for half a minute or more. Then relax your fist, but don’t open it. Delicately slip the fingers out, and then gently try to open your fist. What does it feel like?
2. Stand in a doorway. Push your arms outward, pressing the backs of your hands into the doorframe, hard. Do this for at least half a minute. Then step out of the door, relaxing your arms and shoulders. What happens?
3. Lie on the floor on your back. With your legs straight (knees locked!) have a friend raise one leg and press gently so the leg muscle is stretched. It should be a little uncomfortable (and definitely a little weird) but not painful. Pause for at least half a minute. Close your eyes while your friend lowers your foot slowly back to the ground, holding by the toe. What does it feel like?

The first one in particular is good for moments when you are waiting around or standing in line with a friend. If you know more of these, post them in the comments!

Want to make one? See instructions at the bottom of this post.

What to do with it. Beyond being pretty, they are also hollow. Once you put one together, it’s easy to get inside by removing a piece or two. You can hide things, or use it as the ultimate wrapping paper for smallish gifts.  (Girlfriend Approved!)

Larger Assemblies? Are larger designs possible? Of course, larger sheets of paper make bigger stars; I’m talking about building a shape that uses more pieces and has more points. If you assemble a cube and then a 12-piece star and then a 30-piece star, you will see the pattern of how they become larger and more complex. I took this pattern one step further. After using over 100 pieces, I discovered I had a giant sheet of points that never closes, never becomes a ball. Thus, I learned that larger stars are not possible: the 30-piece design is as grand as it gets.

What does Google think of all this? This is part of the larger world of so-called modular origami. There are many different modules (puzzle pieces) that can build up cool shapes. Michał Kosmulski has a great collection of designs. I don’t see mine there, but his, I think, are generally more intricate.  One uses 210 pieces!

Instructions.

I filmed these videos with an old camera, and my hands seem to move like I’ve downed a dozen espressos. But it gets the job done.

How to make a puzzle piece:
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Folding 30 pieces took me about one and a half hours. Once you’ve got it down, it’s something to do in front of a book or TV.

How pieces fit together to form a point:
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To assemble a piece, just make a point and then build more points off of it. To give you an idea of how it goes, here’s a sped-up video of an entire assembly:
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1. What are the odds of winning a particular dice roll? (like 3 dice vs. 2 dice, 3 dice vs. 1 die, etc.)
2. What are the odds of conquering a territory? (for example, 20 men invading 17 men)

Anyone who has played risk has an intuitive sense of the answers to the first question.  Odds of winning a 3 dice vs. 2 dice battle are about 50/50.  The invading army gets the advantage of an extra die, but ties go to the defender.  It turns out that these advantages roughly balance each other out.  (Full results.)

Question #2 is harder because battles between a large number of soldiers are complicated.  It all comes down to who has to roll with a reduced number of dice.  For example, if a large army is cut down early with a string of bad luck, its odds of winning go down much faster.

 This table gives the odds of winning a whole series of dice rolls and capturing a territory.  The number of invading soldiers is along the side, defending soldiers along the top. Their corresponding entry gives the invaders’ odds of wiping out the defending army. One is always subject to luck, but with this table at least you can know what you’re getting into.  Only the truly dedicated would want to memorize some of this; it is more interesting to look for and internalize patterns. 

Click to enlarge!

About this table:

• Of course there is never a 100% probability of success.  I round to 100% when the probability is greater than 99.5%.
• This table considers up to 20 soldiers, but it could easily be extended.
• It’s obvious that more soldiers = better odds.  But following along the diagonals reveals interesting features. Notice how 2 vs. 1 is better than 3 vs. 2 but not as good as 4 vs. 3.
• To generate this table I computed 3 800 000 simulated conquests (10 000 per entry) using a Monte Carlo algorithm coded in Mathematica.

Previous Work. There is a web article by Daniel C. Taflin (2001) that considers Question #1 and explains the underlying mathematics of his approach in detail.

Other important life lessons. Alliances are made to be broken; Asia is weak; never leave Australia unattended.

Carbon dioxide (CO2) is heavier than most other molecules and elements in the air.  In the same way that Helium rises (think of party balloons), CO2 sinks.

With a very large bowl or container and lots and lots of vinegar and baking soda, you can make a little “puddle” of CO2.  When soap bubbles – which are filled with normal air – hover over this puddle, they float in place like beach balls sitting on top of a pool.  You can see this in the picture, but of course it’s much weirder-looking in person.  Bubbles serenely unmoving in midair.

A Giant Bowl. You need a bowl big enough that the CO2 can slosh around without completely seeping away.

Recipe for CO2. A back-of-the-napkin calculation shows that one big box of Baking Soda is good for about six big bottles of vinegar.  Their reaction produces CO2 gas and a big mess.

Bubbles! The classic recipe is 12 parts water to 1 part blue Dawn dish detergent.  A few tablespoons of glycerin (which can be bought at a pharmacy) help the bubbles last longer.  We tried some variations, like all-glycerin bubbles.  It also seemed like a good occasion for the Bubble Thing!

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Finding the CO2 line. This Four Gas Tester, designed to check if air in a workplace is safe to breathe, showed that the air was normal above the bowl.  But when I dipped the sensor under where the bubbles were floating, red alarm lights indicated that the air was not breathable – too much CO2!

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