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Coins, Prime Numbers, And Checker Boards

Old coins challenge question

(thanks to Alice Loth)

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In the US, with our five coin denominations of 1, 5, 10, 25, and 50, it can take up to eight coins to make change (99). Which different five denominations would minimize the number of coins ever needed to make change?


With coins of 1, 3, 11, 27, and 34, you never need more than five coins to make change. Tenie Remmel reports that this is one of 1,129 solutions, the one requiring the fewest coins on the average: 3.343 coins. Maybe the US should adopt one of these systems, but they all make figuring out how to make change harder, especially since you cannot always use the largest coin possible. Ruth Gato wonders if you can do better if both parties may contribute coins to the change-making process ("you owe 4, give him 7, and he gives you 3.") (Winning answers also from Holly Arrow, Gary De Young, Greg Miller, Chuck Gahr, Robert Lewis, Reed Burkhart.)

Old prime number challenge question

For the new prime number 21,257,787-1, what is the last digit? the second last digit? the first digit? the second digit?


Drake Smith, William Bond, Mike Soskis, and Jan Smith explain why the new prime is 41 ... 27. The whole prime (and other interesting information) is on the World Wide Web at:, but this prime is too big to load into most personal computers, so you have to find another way.

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Fortunately, the last digit of powers of 2 repeats every 4 times, with more certainty than last night's eclipse, and the last two digits repeat every 20 times. Dividing 1,257,787 by 20 leaves a remainder of 7. Since 27-1=127, the last two digits are 27.

Finding the first two digits takes logarithms and a calculator. The log (base 10) of the new prime is just 1,257,787 times log 2, or about 378,631.615. Hence the new prime is about 10378,631.615 = 100.615 x 10378,631, or about 4.12 x 10378,631, a number starting with 41 and having 378,632 digits. Aubrey Dunn points out that in base 2, the new prime is all 1's: 11 ... 11 (1,257,787 of them).

New Checkerboard Challenge

(thanks to Brian Barniett)

How many squares are there on an 8x8 checkerboard, counting not only the 64 1x1 squares, but also the 2x2 squares, 3x3 squares, etc.? What about a 101x101 checkerboard?

How many of those squares contain the little square in the center?

*Send answers, comments, and new questions to:

Math Chat

Bronfman Science Center

Williams College

Williamstown, MA 01267

or by e-mail to:

The best submissions will receive as a prize a copy of the book Flatland, which helps explain higher dimensions. The challenge question answer and winner will be announced in the next column in two weeks.

Geometry of an eclipse

Meanwhile, if you've ever wondered why eclipses, such as last night's lunar eclipse, are so rare, because the moon's orbit is tilted, carrying it out of the plane of the earth's orbit. Otherwise, we would have a lunar eclipse at every full moon and a solar eclipse at every new moon. Lunar eclipses are more common than solar eclipses, because the earth is bigger than the moon and casts a bigger shadow on the moon (lunar eclipse) than the moon casts on the earth (solar eclipse).

Math Chat is a biweekly column, based on the live call-in TV show in Williamstown, Mass.