# Frank Morgan's Math Chat - Old Coins

APRIL 5, 2001

Old Challenge (John George). Each year a kingdom mints one million new coins, and a random ten percent of all old coins are destroyed. About how old do you expect the oldest coin in circulation to be?

Best Response (Joseph DeVincentis): Expect the oldest coin to be about 162 years old. In steady state, the million new coins balance the ten percent destroyed; so the total is ten million. Of a million new coins, expect 90% = .9 to survive one year, .9x.9 to survive two years, .9n to survive n years, and one lone coin to remain when 106x.9n = 1 or n = 131 years, the most common answer from readers. But 131 is just an average, some years are luckier than others, and a coin minted in a lucky year would survive longer. To factor this in, consider all ten million coins, old and new, with a lone survivor expected when 107x.9n = 1 or n = 153 years. This answer is still inadequate, because we still have to add on the age of the remaining coin of the ten million at the beginning of the 153 years. At that time, 10% = .1 of the coins were new, .1x.9 were one year old, .1x.92 were two years old, etc, so that the expected age, obtained by multiplying each possible age by its probability, was

0(.1) + 1(.1)(.9) + 2(. 1)(.92) + . . . ,

which turns out to be 9 years, for a total expected age of 153+9 = 162 years, the best answer we got.

All of these arguments are, however, a bit hueristic, since we considered only one time period (153 years) instead of considering all time periods times their probabiities. Would a rigorous computation yield a similar answer?

Questionable Mathematics. Jonathan Falk found this item in Tony Snow's February 23 column at townhall.com:

"Question: Name one thing beneath Bill Clinton's dignity. Answer: This is a trick question, like asking whether zero is odd or even. It has no known answer."

Zero is even of course.

Readers are invited to submit more examples of questionable mathematics.

New Challenge (Joe Shipman). Larry King said in his USA Today column that there are 293 ways to make change for a dollar. Is this correct? (Assume only currently minted denominations.)

Send answers, comments, and new questions by email to Frank.Morgan@williams.edu, to be eligible for Flatland and other book awards. Winning answers will appear in the next Math Chat. Math Chat appears on the first and third Thursdays of each month. Prof. Morgan's homepage is at www.williams.edu/Mathematics/fmorgan.

THE MATH CHAT BOOK, including a \$1000 Math Chat Book QUEST, questions and answers, and a list of past challenge winners, is now available from the MAA (800-331-1622).