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A Beetle Math Puzzleby rjlipton Thursday February 20th, 2020 at 8:42 AM

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Lessons from a puzzle about prime numbers

 [ Wikipedia ]

Doron Zeilberger is a famous combinatorial mathematician based at Rutgers. He is noted for actively using computers in research. His computers even get co-authorship credit under the name “Shalosh B. Ekhad,” which is Hebrew for 3B1—a computer that came from building 3, corridor B, room 1 of AT&T Bell Labs.

Today I thought we would talk about a recent joint paper of Zeilberger on Covering Systems.

This paper has one co-author who is human, Anthony Zaleski, also of Rutgers. It starts with a puzzle about beetles on a circular track. The puzzle does not need a computer to solve—though a computerized visualization would make it more enjoyable. It makes several interesting points, points that are distinctly human, and I hope you might enjoy it.

The Puzzle

They ascribe the puzzle to Jean-Paul Delahaye, who modified Peter Winkler’s writeup of a folk puzzle that Winkler stated about ants.

One places nine beetles on a circular track, where the nine arc distances, measured in meters, between two consecutive beetles are the first nine prime numbers, ${2,3,5,7,11,13,17,19,23}$. The order is arbitrary, and each number appears exactly once as a distance.

At the starting time, each beetle decides randomly whether she would go, traveling at a speed of 1 meter per minute, clockwise or counter-clockwise. When two beetles bump into each other, they immediately do a “U-turn”, i.e. reverse direction. We assume that the size of the beetles is negligible. At the end of ${50}$ minutes, after many collisions, one notices the distances between the new positions of the beetles.

Note that there are two levels of probability: the initial order of the vector of distances and the initial direction of each beetle. Yet after ${50}$ minutes there is a high probability of the distances being exactly the same: the first nine prime numbers. Is this a miracle? What is the probability?

The Lessons

The point is we have deliberately stated the puzzle to make it harder. The way we stated it is misleading and the following lessons are hints to help solve the puzzle.

• That the arc lengths are prime numbers is unimportant. The puzzle works whether or not the distances between the beetles are prime numbers.

• The probability is ${1}$ that the beetles are at the given positions. There is no chance that they will not arrive at the antipode points. None.

The Solution

The first observation is that the circular track’s length is

$\displaystyle 2+3+5+7+11+13+17+19+23=100.$

The second is that the probability is ${1}$ that the distances are the same after ${50}$ minutes. Imagine that each beetle carries a flag. Instead of reversing direction when they collide, let the beetles exchange their flags and continue moving as before. Now the flags always are moving in the same direction at the same speed. This means that after ${50}$ minutes the flags are at the antipode position. But the beetles are located at the same places as flags, and so the distances are the same as before. Note, the beetles are each located at the position of some unique flag, but which flag can change many times.

The only constraint is that the distance traveled in ${50}$ minutes divides the length of the circular track. The fact that the distances are primes is never used.

Open Problems

Zaleski and Zeilberger say:

We point out that very often primes are red herrings. This is definitely the case for covering system, and who knows, perhaps also for the Riemann Hypothesis.

I assume this is a bit tongue in cheek, but their point is valid. Do we miss solutions to problems when we use information that is not really important? How do we decide which information is key and which is redundant? This why I like this puzzle.

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jepler
12 minutes ago
Earth, Sol system, Western spiral arm

Can You Solve the 'Hanging Cable' Problem, Used as an Amazon Interview Question? Wednesday February 19th, 2020 at 9:46 PM

An anonymous reader shares a problem that Amazon asks in its interviews: A cable of 80 meters is hanging from the top of two poles that are both 50 meters off the ground. What is the distance between the two poles (to one decimal point) if the center cable is (a) 20 meters off the ground and (b) 10 meters off the ground?
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jepler
11 hours ago
no, I couldn't do it on the spot. your favorite interview question is dumb.
Earth, Sol system, Western spiral arm
1 public comment
zwol
14 minutes ago
This is like the company once heard of Fermi problems and has been playing telephone with the concept.

Regardless, if someone sprang this on me I would say “I would have to look up the formula for that, shall I demonstrate my ability to do so?” and if they didn’t accept that I would walk out.
Pittsburgh, PA

What to do with underground parkingby Gordon Price Wednesday February 19th, 2020 at 8:27 AM

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Ian Robertson found one solution in Paris.  From Euroactiv:

In Paris, as in many European cities, the number of cars is declining, which is leaving a vast amount of underground car parks empty. With its start-up project called “La Caverne”, Cycloponics is reclaiming these urban territories and using them as a way of growing plenty of organic vegetables. …

At Porte de la Chapelle in Paris, the two have set up a 3,500 m2 urban farm located underground, in a former car park. …  Gertz and Champagnat responded to call for tenders from Paris, whose empty car parks were squatted by consumers and crack dealers. It’s been more than two years now since ‘organic has replaced crack’, and about fifteen jobs have been created. …

Small packets of water-soluble, sterilised and packaged straw are hung from floor to ceiling, and the mushrooms grow through tiny holes. Everything is calculated to ensure their optimal growth. The air is saturated with moisture, the endives grow in the dark, and the mushrooms get a few LED lights.

But the car park has definite advantages over the limestone cavities usually used to grow mushrooms, as there is a permanent and precise control of the weather, as well as better thermal stability. …  Farming in car parks also makes it possible to better resist the climate crisis. Parasites and other insects, for instance, are rather rare in the subsoil, even if endive tubers and straw bought outside can also be vectors of diseases, such as sclerotinia, which destroyed part of this year’s endive harvest. …

“In Paris, as in many European capitals, people no longer have cars, there are too many parking lots, especially in the poorest districts. But we also visited unused car parks on the Champs-Elysée. It would be possible to do something about it!” according to the entrepreneurs.

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jepler
1 day ago
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iridesce
1 day ago
DC

Sherbet Custom Game Pad is Smooth and Sweetby Kristina Panos Monday February 17th, 2020 at 6:53 PM

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When [Billiam]’s beloved Logitech G13 game pad went to that great spectate room in the sky, he decided to pay homage by designing a custom, more ergonomic replacement from the desk up. Grab a spoon and dig into the story of [Billiam]’s journey toward Sherbet, because it’s a sweet ride.

Here’s the scoop: like a lot of DIY game pads and keyboards, Sherbet is based on the Teensy. We often see the micro USB coming straight off the Teensy, especially in clear acrylic builds, but [Billiam] added a USB breakout board so there’s no direct stress on the Teensy itself.

One of [Billiam]’s design challenges comes from the game pad placement — he has a tall desk and uses a keyboard tray, so it has to fit the space and leave enough room for his hand. Fortunately, there are low-profile mechanical switches out there, although the keycap options are strongly limited. We love that [Billiam] embedded a tiny ceramic bearing into one of them to use as a homing bump, because that’s a great idea.

If you want to take a crack at this project, [Billiam] has all the goodies laid out. [Billiam] wanted to use QMK firmware, but they didn’t have joystick support yet, so he’s got an Arduino sketch running in the meantime.

We love a good custom game pad around here, especially if they can reprogram themselves.

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jepler
2 days ago
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Why Do Philosophy? Monday February 17th, 2020 at 6:50 PM

2 Comments and 9 Shares
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tante
2 days ago
"Why do philosophy?"

It's a fun comic though it forgets that philosophy itself is mostly white dudes arguing the rules of made-up games.
Berlin/Germany
1 day ago
I think that was partially addressed by the alt-text.
codersquid
1 day ago
Subscribe to examinedworlds.blogspot.com. It's an sf blog by a philosophy prof who teaches non-western philosophy. (he's got some cool posts about non western vs western (old white dudes) philosophy)
jepler
2 days ago
Earth, Sol system, Western spiral arm

Making radioactive uranium glassby NileRed Saturday February 15th, 2020 at 9:27 PM

1 Comment
From: NileRed
Duration: 30:03

Visit https://Brilliant.org/NileRed to get 20% off a year of Brilliant Premium!

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For this project, I'll be making radioactive uranium glass, which was very popular 100 years ago. Under a blacklight, the uranium in it fluoresces to give off a nice green color.

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Nile Red Shop: https://nile.red

Patreon: https://goo.gl/3h353G

Instagram: https://www.instagram.com/nile.red/

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jepler
4 days ago
uranium
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