Thursday, June 7, 2012

3 3 7 7

Here's a fun puzzle (thank you Piotr!):

Using only multiplication, addition, subtraction, division, and parentheses, combine all of the four numbers 3, 3, 7, and 7 to get 24.

Apparently this is an example of the "24" game in which you do this with any other four numbers. (A related game I remember from middle school is to use the four digits of the current year to make as many numbers from 1 to 100 as possible, using more creative operations like !, mod, and exponentiation. This worked better in the 90s :)

This is a fun one in particular because it's really hard to figure out in your head, but when you use pencil and paper to explore more systematically, the two double numbers cuts down the number of possibilities drastically and it's very doable.



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Now don't keep reading until you figure it out!



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The answer is to use fractions in the intermediate values in an unobvious way. Three sevenths plus three is twenty four sevenths. And there you go. (I spelled that out in words intentionally because you can't glance at it accidentally and give away the answer to yourself.)

Wednesday, March 14, 2012

happy pi day!

So I got to say on the radio for pi day that pi is the circumference divided by the diameter of a circle. And  short story shorter, this got me thinking about why people like pi so much in the first place. I think it comes down to a combination of three things:

First, pi shows up everywhere in science. From the time you learn basic geometry in elementary school, pi is popping up in your school notebooks. Then you move on to trigonometry, calculus, Fourier analysis, and on up as complicated as you want to get, and that pi is carried along for the whole ride. Anything oscillatory turns out to be described with sine and cosine functions, which boil down to geometry of circles, so waves, pendulums, light, sound, planets, optics, electrical currents, et cetera et cetera, all involve pi. Then it turns out that that other famous constant, e, is also related to pi, so population growth, electric charge, compound interest, probability distributions, and anything else exponential in nature, also all contain pis lurking quietly in the background. And somehow, even when you get into the domains of pure mathematics that seem superficially disconnected from all of that other real-world stuff, pi keeps showing up. The sum of the reciprocal of each natural number squared? There's a pi in that. Is it any wonder that pi begins to feel like a familiar friend?

But that's not all, of course. Lots of numbers appear all over the place. 10. 2. Physicists have the speed of light. Chemists have Avogadro's number. Why don't these constants have the same appeal as pi? Unlike these other boring old numbers, pi is shrouded in mystery as a result of being irrational and transcendental. Its irrationality means that you can't write it down as a fraction, and that if you try to write down its digits, the sequence will continue forever without repeating. Transcendence is like turbo-charged irrationality. Not only can you not write it down as a fraction, you can't calculate it with any combination of whole numbers and algebraic operations like division and exponents and roots. You can get closer and closer the longer you try, but you can never quite get there. There's nothing to do but symbolize it with a Greek letter and forget about the fact that you can never be exactly sure what it represents.

But that's still not the whole story. e is also transcendental. The square root of 2 is irrational. i is certainly mysterious in its own way. There are plenty of other loved constants, but still none approach the popularity of pi. And that comes down to pure nostalgia. For many of us, learning about pi is the first time we catch a glimpse of the immense mystery of the universe and realize we can't hold on to it, put it in a box, and study it in all thoroughness. We have to content ourselves with squinting at it from many angles and then try our hardest to put together a coherent abstract picture in our minds. The mystery never ends, and for us mystery-junkies, the scientists, that first glimpse into the infinite abyss is an unforgettable moment that continues to drive us throughout our lives. And even after we come to terms with all our numerical friend implies, each time we casually say hello, a tiny part of our subconscious mind is reminded of that deliciousness of discovery.

Sunday, February 26, 2012

symmetry is awesome

I heard this cute math puzzle recently (thanks Piotr :)

Three ants start at the vertices of a unit equilateral triangle. Ant 1 always faces ant 2, 2 always faces 3, and 3 always faces 1. They all start walking at the same time towards their target neighbor, at one unit per second. What happens?

Now I'll wait while you figure that one out.

In the meantime, here's another nice symmetry-related thing that Joanna showed me: The surface area on a sphere in between any two parallel slices through the sphere is only dependent on the vertical distance between the slices. So if you slice off the top mile of the Earth, and take out a one mile slice at the equator, those sections have the same surface area.

That's a little surprising, until you realize that since sphere are perfectly symmetric (they have constant curvature in any direction), the circumference of the slice you're taking is shrinking as you move away from the equator at exactly the same rate that it's flattening out, so to speak. That is, three inches "above" (3 inches along the axis of rotation) the equator translates to three inches on the ground, but three inches "below" the north pole translates to a looong way on the ground. And this exactly offsets the fact that the Earth is much bigger around at the equator than the pole.

...(ant spoiler alert)...

Ok back to the ants. By symmetry, they will collide at the center of the triangle. The question is, how long does it take for them to get there? They're spiraling inwards as they turn to keep facing each other, so they don't take a straight route there. But, by symmetry, at every point in time they're still in an equilateral triangle formation, essentially starting from scratch on a slightly smaller and rotated triangle! That means that if you figure out what's going on at the first instant, the same thing applies at every later instant.

From there it's easy. The inward distance from each vertex to the center of the triangle is √3/3. And the component of each ant's velocity that is pointing inward is cos(30)=√3/2. So they will collide in the center in 2/3 of a second.

Update: arithmetic corrected... again.

Thursday, February 23, 2012

Mercury and a 28-hour moon

Yesterday there was an aesthetically nice astronomical coincidence. Even better, I got to observe it from the comfort and convenience of the economics grad student lounge, which has a beautiful view towards the west from the 6th floor of a building up a hill in Berkeley. This means you can see all the way to the western horizon, so you can observe objects that fall right on the cusp of possibility, setting just after the sun itself.

Today Mercury and a 28-hour old moon coincided in this brief window. Mercury by itself is very difficult to observe: since it orbits close to the sun, you can only see it just after sunset or just before sunrise, and even then only rarely, when it's as far away from the sun as it ever appears to us (with the Earth and Mercury forming a right angle with the sun.) The last time I saw it definitively was in 2001 (although I can't say I've often tried; I'm more of a deep sky fan than a solar system buff...)

The same applies to extremely thin crescent moons. A goal of any lunar observer is to pick out the hairline crescent as soon as possible either before or after the new moon, when it passes directly between us and the sun and is momentarily invisible. 28 hours old is definitely not the youngest moon you can observe, but I think it's my personal record, and is definitely sufficiently strikingly beautiful.

So, some pictures:

Mercury and crescent moon side by side over bay, shortly after sunset (about half an hour)

moon through binocular eyepiece

Mercury and moon a little later, as it's getting dark

Jupiter (upper left), Venus (left of middle) and moon sliver (on horizon). Unfortunately Mercury didn't quite get picked up in this exposure, but you could still see it naked eye at the time.