Katherine Nash Scafide, who is studying nursing with a specialty in forensics, recruited me to help her measure the impact force of a paintball. Her work focuses on the bruising, but for my part we didn’t use any human targets. We wanted to know how much firing distance mattered — how differently a shot from 15 feet and a shot from 30 feet might arrive at the target. (Since this was just an exercise to gather background information, she generously granted me permission to share it.)

Her husband took shots at a target next to my lab’s high-speed camera. We recorded shots from several distances, and I analyzed the videos to measure the paintball’s incoming velocity.

In most cases, the paintball completely smashed onto the target without backward splashing or flying bits. (The video above was an exception, a neat-looking one.) The impact occurred during one or two frames of high-speed video, meaning 0.0005 or 0.001 seconds. From this and the paintball’s mass (3.2 grams) we can get an estimate of impact force. Fired at 15 feet, the average force during impact is in the range of 250-500 Newtons, or 45-90 pounds. Ouch.

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With a trip to the toy store and an hour of snapping and unsnapping LEGOs, I came up with a little gearbox connecting a motor to a stirring rod. The gears give this light-weight motor the mechanical advantage that it needs to churn thickening ice cream. For each turn of the rod, the motor spins 15 times.

The assembly — shown with some spare pieces attached as stirring implements — fits snugly into a large empty yogurt container with notches cut in the sides.

Ice cream is milk : cream : sugar in the proportion 2 : 2 : 1 with a pinch of salt and flavoring. Just mix them and agitate the mixture while it freezes. (Other, more complicated recipes involve heating the milk and using eggs too. No need for more complication here.)

I let it chill in peace for about an hour. At 7:30, when I could see ice crystals starting to form, I switched on the motor, headed out the door for the evening, and hoped for the best. Around midnight, I came home to a delicious success!

Thanks to Jeff Potter for sharing the idea of LEGO ice cream makers in his book. With the help of the enthusiastic people at Shananigans Toy Shop, I decided on LEGO Power Functions (#8293) for a motor and LEGO Crazy Action Contraptions (Klutz Press) for a starter set of bricks. It was just the right number of pieces to give me room to experiment and create something sturdy enough. Try making one yourself!

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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.

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This is a screenshort. Click it open the BPM calculator.

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

What to do with it. All of these are 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. What happens if you try to use even more pieces? After experimenting with more than 100 pieces, I realized that I had a giant sheet of points that never curves to close on itself. It never becomes a ball. Larger stars in this kind of pattern 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. The video is jittery, 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!

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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1. Get a medium-sized plastic soda bottle and a party balloon.
2. Puncture a hole in the side of the bottle near the bottom.  It’s easy if you use a nail heated with a match or a candle.
3. Dangle the balloon inside the bottle and open it over the bottle’s lip.  (See photo.)
4. Inflate the balloon inside the bottle.
5. With your mouth still on the bottle, place a finger snugly over the hole in the side.
6. The balloon stays open on it’s own!  Fill it with water from a faucet.
7. Bring it outside, and take your finger off of the hole…

Science! To quote Zap Science (see section “Soda Bottle Science”), we live at the bottom of an ocean of air.  Air pushes in upon everything.  The weight of the atmosphere pushes down and outward on the balloon but, when the hole is covered, there is not much air pushing back.  The pressure of the atmosphere – all the air piled up over your head – is working to hold the balloon open.  When you uncover the hole, you let the atmosphere back in (so to speak) and – splash! – the balloon is free to shrink again.

Stray tips. It’s best if you use a normal rubber party balloon, not a small water balloon.  Also, 1-liter bottles are best.  2-liters tend to fold in on themselves to easily under pressure.

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