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# 486

The Mathematician's Lottery - March 1, 2012
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Nothing original from me today, but a comic based on a well known joke!

For those who haven't learned about the harmonic series yet, this is the summation:

This well known series diverges to infinity and thus, after a finite number of terms will have a sum of 750 trillion.

Problem 1: Determine how many years it will take for the "losers" to get their money back (50 euro). [[Is it 50 euro or 50 euros?]]

Problem 2: Determine how many years it will take for the lottery to pay out in entirety, i.e., 750 trillion!

For one, there will be rounding issues. Shall we round to the nearest cent? Or round up to the next cent?

Perhaps pay out in Bitcoins; then you have 8 decimal places.

Using some guessing and Wolfram Alpha, I got 2.911×10^21 weeks or 5.583×10^19 years (approximately 4.1×10^9 × the age of the universe) for problem 1.

2,911,002,088,526,872,100,231 weeks to be more exact.

An estimate for the nth harmonic number is
h = ln n + .5772
(This is fairly poor at first but at large n it is plenty good for 4 significant figures.)

The inverse of this is
n = 0.561468 e^h

setting h = 500 trillion, we get n as approximately
2.381 x 10 ^ 217147240951 weeks.

This is about 4.566 x 10 ^ 217147240949 years

or 3.3 x 10 ^ 217147240939 times the age of the universe

(Thanks to mrob's hypercalc.pl large number calculator, and Wolfram Alpha for converting weeks to years and universe ages. No, WA can't handle numbers that large; but this gave me the mantissa, and I could see how the exponents decreased.)

This reminds me of the _______ (fill with your favorite group to pick on) Lottery: it pays out \$1 for a million weeks when you win!

According to dictionary "euro" and "euros" is correct. The plural "euro" is European i think (since it's the same in other European languages).

Well, I'm spanish and we say "Euros". In France they say "Euro", and in Great Britain, Euros.

Great comic!

I think you're "supposed" to say euro. Btw, Mike, are you Irish? Because that's the only English-speaking country that uses the euro so I'm curious...

I'm Canadian and so I'm only used to saying dollars! I figured I'd mix things up a bit for this comic though.

i would demand that you would stick to your payment schedule, and then take the cut the bank takes from each transaction...

Put Inflation in btw and then Boom!!

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Hello my fellow math geeks. My name is Mike and I am the creator of Spiked Math Comics, a math comic dedicated to humor, educate and entertain the geek in you. Beware though, there might be some math involved :D

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