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14 Mar 2007
Famed Physicist Richard Feynman once joked that he wanted to memorize π up to the 762nd digit. Why? Because that's where pareidolia kicks in, and the digits appear to briefly coalesce into rationality:
3.1415926535 8979323846 … [727 digits] … 9605187072 1134999999 …He would end a hypothetical recitation at that point, the implication being that from there on the decimal repeats. Virtually everyone, obviously including Feynman, knows π is both irrational and transcendental, and thus such a fact is impossible. The so-called "Feynman Point" is the first incidence of six consecutive identical digits in π and it also happens to be the first incidence of four and five consecutive digits.
Happy Pi Day.
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28 Jul 2006
Not many people know that there is very little copper in modern pennies. In 1982, they were changed from 95% copper to simply copper-plated zinc. If you cut open a penny made after that date, you'd see the grey-colored metal on the inside. In fact, in 1981, pennies of both recipes were made, and collectors have to listen to the sound they make when they bounce to tell the difference. Do it yourself with a penny from the 70s and one from the 90s; the difference will be obvious.
The price of copper closed last night at $3.48 a pound. That means that a 3.1 gram penny has about 2.26 cents of copper in it. Melt down a big wad of those, and you can make yourself a tidy profit. The price of zinc has been escalating in the last few months, too, so it might not be long until it's worthwhile to melt down new pennies, too.
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14 Mar 2006
Pi Day is a sham. So-called Pi Approximation Day (July 22, or 22/7 in European date style) is closer to the true value of π.
22/7 = 3.142857142...
abs( 22/7 - π ) = 0.001264489...
abs( 3.14 - π ) = 0.001592653...What a shame.
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15 Feb 2006
Every time the Powerball Lottery breaks two-hundred million or so, people at work start talking about it and emailing appeals for joining a lotto pool. After I got sick of shaking my head at these people, I realized that it could possibly be worth it to buy a ticket. Since the ticket price stays the same even as the jackpot grows, there's got to be a tipping point, beyond which the payoff from the average ticket is greater than one dollar. But what is that point? Should I be queuing up at the gas station right now? Or should I wait until it hits ten digits?
Powerball tells you that the odds of getting the grand prize are 1 in 146,107,962. This gives us a good starting point. In a universe without taxes, annuties or interest, a rational person wouldn't want to purchase a ticket until the jackpot was $146 million. But we don't live in that universe, nor do I believe we'll be travelling to it to play the lottery.
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11 Aug 2005
I love math. Not just complex math, but arithmetic, too. I simply adore mathematical shortcuts. That's why I spent the last couple of days finding shortcuts for divisibility tests:
- 3 - Sum of digits is divisible by 3.
- 4 - If the last two digits (tens and ones places) are divisible by 4. You can simplify this test by subtracting multiples of 2 from the tens place.
- 7 - Remove the digit in the ones place. Double it, and subtract that from the remaining number. Repeat until you have a one-digit number. If that number is 7, 0, or -7, the original number is divisible by 7.
- 8 - If the last three digits are divisible by 8. You can simplify this test by subtracting multiples of 2 from the hundreds place, and multiples of 4 from the tens place.
- 9 - Sum of digits is divisible by 9.
- 11 - If the alternating sum of the digits (first digit, minus second digit, plus third digit, etc) is divisible by 11.
- 13 - Remove the digit in the ones place. Multiply that digit by 4, and add that to the remaining number. Repeat until you have a number less than 40. If that number is 13, 26, or 39, the original number is divisible by 13.
The rules for 7 and 13 can easily be extended to just about any number under 100 (and, in fact, the rules listed here for 3, 9, and 11 are simple variants). As long as you understand how those ones work, you don't even really need to memorize anything.
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15 Jun 2005
Nomad refuses to carry coinage. I recall one time (likely apocryphal) he purchased a soda for $1.25 at a train station, and then put his change (all seventy-five cents) on top of a garbage can and left it. I can't remember if I did the sterotypical thing and picked it up, but I'm sure I called it foolish.
But it got me thinking; is carrying change a waste of energy? If you carried a pocket-full of coins all day, could you purchase more calories of food than you burned carrying it? I thought about it for a while, and I couldn't reason it out, so I decided to calculate it. The answer somewhat surprised me.
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22 Feb 2005
A new trigonometry challenge: Given a hexagon inscribed in a square as shown below, what is the
ratio ofrelationship between the length of segment a and segment b? Assume the hexagon's sides are all of equal length.
Update Feb 26: The solution, props to Brian.
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10 Jan 2005
An offhand reference in The Golden Ratio led me to research Archimedean solids, which led me to a page with Paper Models of Polyhedra. Fun!
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22 Nov 2004
IBM's monthly teaser, Ponder This, seems like fun. Part 1 of November's only took me about 5 minutes and 66 characters of Perl. Part 2 is a little more analytical. I'll work on it at lunch.
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3 Nov 2004
American paper sizes, like all other Imperial forms of measurement, are so arbitrary. It's quite a breath of fresh air -- a veritable gale of freshness, in fact -- to read about the mathematically sound basis of the rest of the world's paper sizes. Bone up on your geometric averages and square roots, and jump into the wonderful world of the international standard paper sizes.
