PRINCETON, NJ — Which Three States Meet Three Times?
Old challenge (Mathematical Diary of 1825)
"Ten pounds [sterling] a quarter are allowed to five auditors, A, B, C, D, E, of a Fire Office. They are required to attend seven times in the quarter, and the absentees' shares are to be divided equally among such as attend.
Now A and B never fail to attend, C and D are each absent twice, and E is absent only once: What is each auditor's share of the given sum?"
The best short answer is, "It depends." It depends on whether C, D, and E are absent at the same time and their shares go just to A and B, or whether they alternate and profit from each other's absence.
Peter Manos considered the first case and computed that A and B receive 110/21 pounds or about 5 1/4 pounds each, C and D receive 30/21 pounds each, and E receives 40/21 pounds. Chuck Medler and Joe Shipman worked out all seven possible cases.
Michael Marcotty, Keith Schleiffer, and Bill Hasek may have found the nicest alternative solution. Marcotty figured, "There were a total of 30 attendances out of a possible 35. Therefore the auditors should be paid one 30th of the 10 pounds for each attendance. These were pounds sterling before decimalization, therefore there are 20 shillings in the pound and 12 pence in the shilling and each attendance should be rewarded by 6 shillings and 8 pence."
Thus A and B receive 2 pounds, 6 shillings, 8 pence; C and D receive 1 pound, 13 shillings, 4 pence; E receives 2 pounds. Total: 10 pounds.
Schleiffer admitted, "First pass through, I used only two decimal places and was forced to deal with half-pence left over, which I allocated to the poor-box. This was, after all, Victorian England."
My own Williams College lies in Williamstown, in the extreme northwest corner of Massachusetts, where Massachusetts, New York, and Vermont all meet at a point. Maryland, Virginia, and the District of Columbia all meet at two separate points.
Can you name three US states that all meet at three different points?
New weekly TV series
PBS has a new TV series on mathematics called "Life by the Numbers," hosted by Danny Glover. Check local listings and the Web site at www.mathlife.com.
Learning to enjoy math
A question about a teenager who is good at everything except math appeared in a recent "Ask Marilyn" column. Marilyn recommended concentrating on the student's strengths.
I agree, but I would also encourage finding something related to math that the student could enjoy, whether or not it has anything to do with schoolwork, maybe something from Math Chat!
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