Saturday Puzzle #25 – Penny For Your Thoughts

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Imagine a very wealthy and eccentric friend (which is the best kind of friend to have) offers you the following choice:

  • One penny on the first day of January, two cents on the second day, four cents on the third day, and so on, doubling the amount you receive each day up to the 31st day of January.
  • One million dollars

Which option would you choose?

Solution: Today’s problem illustrates the power of a geometric series. It starts out very slowly, 1 cent, 2 cents, 4 cents…it seems like child’s play but by the end of the month, watch out! The number of pennies you receive on day N is given by 2 raised to the power N-1, which we can write mathematically like this: f(x) = 2**(x-1). The following graph of this function illustrates the sudden, rapid growth of a geometric series:

On the last day of January you would receive 2**(N-1) = 2**30 or 1,073,741,824 (over one billion) pennies, which is more than 10 million dollars! But it gets even better because we have to sum the pennies received throughout the entire month of January. The sum of all pennies received through day N is given by (2**N) – 1, which, in our case, would be (2**31)-1. That comes to 2,147,483,647 cents or, roughly, 21.5 million dollars.

Congrats to Simon Banks, Morag Livingston, Mudassir Ansari and Neal Starkman, all of whom wisely chose the pennies. Neal also pointed out the need for penny wrappers and hired help, which presumably can be paid with a portion of your $20M windfall.

4 thoughts on “Saturday Puzzle #25 – Penny For Your Thoughts”

  1. “A fool and is money are soon parted”
    So a binary 1 which has a 0 added to it for 31 days..
    Expressed in binary this would be 1000000000000000000000000000000
    In C/C++ this is expressed as 1 << 31
    Expressed in decimal this would be 2147483648

    So I would take the 2.1 trillion please

  2. Howdy! I would most certainly taking the cents – assuming you can keep every day’s worth you’d get more than twenty times as much by 31 January – you’d be a multi millionaire… $21,474,836.48 to be precise!

    I don’t suppose by any stretch of the imagination you could be my rich uncle!!!

  3. I will opt for the first option, as this is a Geometric series, which is calculated by the formula a(1-r^(n+1))/1-r, where a=1, n=30, r=2, which comes out to be $21,474,836.47

  4. I would choose second option in case my friend changes his mind. As most of money comes from last days it is very risky to choose first option :)

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