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23 November, 15:17

A leprechaun places a magic penny under a girl's pillow. The next night there are 2 magic pennies under her pillow. The following morning she finds 4 pennies. Apparently while she sleeps each penny turns into two magic pennies. The total number of pennies seen under the pillow each day is the grand total; that is, the pennies from each of the previous days are not being stored away until more pennies magically appear. How many days would elapse before she has a total of more than $2 Billion?

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  1. 23 November, 15:40
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    32 days

    Explanation:

    Simulate (build a table) the growing of the number of pennies for some nights to figure out the pattern:

    First night: 1 penny = 2⁰

    Second night: 1 * 2 pennies = 2¹

    Third night: 2 * 2 = 2²

    Fourth nigth: 2² * 2 = 2³

    nth night: 2ⁿ⁻¹

    You want 2ⁿ⁻¹ ≥ 2,000,000,000

    Which you solve in this way:

    2ⁿ⁻¹ ≥ 2,000,000,000

    2ⁿ⁻¹ ≥ 2,000,000,000

    n-1 log (2) ≥ log (2,000,000,000)

    n - 1 ≥ log (2,000,000,000) / log (2)

    n - 1 ≥ 30.9

    n ≥ 31.9

    Since n is number of days, it is an integer number, so n ≥ 32.

    Hence, she will have a total of more than $ 2 billion after 32 days.

    You can prove that by calculating 2³² = 2,147,483,648.
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