YouTube // Numberphile
YouTube // Numberphile

The Puzzling Collatz Conjecture

YouTube // Numberphile
YouTube // Numberphile

The Collatz Conjecture is a relatively simple set of math instructions that lead to a puzzling problem. If you run this set of rules on a given number, and repeat the process, where do you end up? In every case that mathematicians have tried since the problem was first posed in 1937, they've ended up at the number 1, but the experts can't prove that this will be the case for all (positive, whole) numbers. Why not?

Here's the sequence: Pick a number that is a positive integer. (For instance, the number 1 or 100 or 10,123,456.) If it's even, divide it by two. If it's odd, multiply it by three and add one. Take the resulting number and keep running the process.

In this video, professor David Eisenbud runs the number 7 through this process and ends up at 1. At present, mathematicians have run all whole numbers up to 2^60 through this process and they all end up at 1. But the tricky bit is that the path back to 1 is often winding and bizarre, not following an obvious pattern. Why? This is genuinely surprising:

If that's not enough for you, here's another six minutes of footage on the same topic:

See also: this highly relevant xkcd comic about the Collatz Conjecture.

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Honey Bees Can Understand the Concept of Zero
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The concept of zero—less than one, nothing, nada—is deceptively complex. The first placeholder zero dates back to around 300 BCE, and the notion didn’t make its way to Western Europe until the 12th century. It takes children until preschool to wrap their brains around the concept. But scientists in Australia recently discovered a new animal capable of understanding zero: the honey bee. According to Vox, a new study finds that the insects can be taught the concept of nothing.

A few other animals can understand zero, according to current research. Dolphins, parrots, and monkeys can all understand the difference between something and nothing, but honey bees are the first insects proven to be able to do it.

The new study, published in the journal Science, finds that honey bees can rank quantities based on “greater than” and “less than,” and can understand that nothing is less than one.

Left: A photo of a bee choosing between images with black dots on them. Right: an illustration of a bee choosing the image with fewer dots
© Scarlett Howard & Aurore Avarguès-Weber

The researchers trained bees to identify images in the lab that showed the fewest number of elements (in this case, dots). If they chose the image with the fewest circles from a set, they received sweetened water, whereas if they chose another image, they received bitter quinine.

Once the insects got that concept down, the researchers introduced another challenge: The bees had to choose between a blank image and one with dots on it. More than 60 percent of the time, the insects were successfully able to extrapolate that if they needed to choose the fewest dots between an image with a few dots and an image with no dots at all, no dots was the correct answer. They could grasp the concept that nothing can still be a numerical quantity.

It’s not entirely surprising that bees are capable of such feats of intelligence. We already know that they can count, teach each other skills, communicate via the “waggle dance,” and think abstractly. This is just more evidence that bees are strikingly intelligent creatures, despite the fact that their insect brains look nothing like our own.

Considering how far apart bees and primates are on the evolutionary tree, and how different their brains are from ours—they have fewer than 1 million neurons, while we have about 86 billion—this finding raises a lot of new questions about the neural basis of understanding numbers, and will no doubt lead to further research on how the brain processes concepts like zero.

[h/t Vox]

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Can You Solve This Ice Cream Cone Riddle?
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iStock

How much is an ice cream cone worth? In this visual riddle by Budapest-based artist Gergely Dudás (who posts comics on Dudolf.com), the answer requires a little math.

The riddle asks you to determine how much an ice cream cone, a scoop of white-colored ice cream (let’s call it vanilla), and a scoop of pink-colored ice cream (let’s call it strawberry) are worth, according to the logic of the puzzle.

Stare at the equations for a while, then scroll down for the answer.

A math riddle that asks you to figure out what numbers each ice cream cone or scoop represents
Gergely Dudás

Ready?

Are you sure?

OK, let's walk through this.

Three ice cream cones multiplied together are equal to the number 27. Since 3 multiplied by 3 multiplied by 3 equals 27, each cone must be equal to 3.

Moving on to the next row, two ice cream cones each topped with a scoop of vanilla ice cream added together equal 10. So since each cone equals 3, the vanilla scoops must each equal 2. (In other words, 3 plus 3 plus 2 plus 2 equals 10.)

Now, a double scoop of vanilla on a cone plus a single scoop of strawberry on a cone equals 11. So if a double-scoop of vanilla equals 4 (2 plus 2) and each cone is equal to 3, the strawberry scoop must equal 1. (Because 4 plus 6 equals 10, plus 1 for the strawberry scoop equals 11.)

And finally, one vanilla scoop on a cone, plus one empty cone, plus a double-scoop of strawberry and a single scoop of vanilla on a cone, all together equals 15. One scoop of vanilla on a cone is equal to 5 (2 plus 3), and an empty cone is equal to 3. Two strawberry scoops plus one vanilla scoop plus one cone can be calculated as 1 plus 1 plus 2 plus 3 (which comes out to 7). So together, one vanilla scoop (5) plus one cone (3) plus a triple scoop with two strawberries and one vanilla on a cone (7) equals 15.

And there you have it.

A cartoon-style legend that shows that one cone equals 3, one white scoop equals 2, and one pink scoop equals 1.
Gergely Dudás

If frozen dairy-themed challenges are your thing, he also has a hidden image puzzle that challenges you to find the lollipop in a field of ice cream cones. Check out more of his work on his website and Facebook.

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