Our cleaning lady Bessie is good with numbers. I mean really good
with numbers. The daughter of a New York City cabbie and
granddaughter of a sharecropper, her humble origins belie her
mathematical genius.
One day she came into my office to do the cleaning and saw me
working on the latest cache I was about to place. When she looked
at the coordinates on the worksheet, she asked what they were. I
explained geocaching and the coordinates, pointing out the degrees,
minutes, and decimal minutes format used by geocachers.
"Now that's interesting," Bessie remarked. "The last four digits of
the latitude are the same as the number on my daddy's taxi. When he
died the cab was taken out of service. He left it to me, but I had
to sell it 'cuz I ain't no cabbie and don't want to live in New
York."
I told her I was surprised that she remembered that rather
unremarkable number.
"Oh, that's a very interesting number," she corrected me. "That's
the lowest number that can be expressed as the the sum of the cubes
of two different positive integers in TWO different ways. Not only
that, the longitude you have there - if you add 10 to them decimal
minutes the last four digits are the same last four digits of the
lowest number that can be expressed as the sum of the cubes of two
positive integers in FOUR different ways. That's really quite a set
of coordinates you have."
I used to be a math major once upon a time, but I couldn't follow.
"You lost me there, Bessie." So she wrote out the numbers to help
me along. This is what she wrote:
N³ + M³ =
ABCD
P³ + Q³ = ABCD
The numbers she wrote where I have put M, N, P, and Q were all
positive integers, and each was different. Sure enough, I checked
her math and it all added up. The last four digits of the latitude
of my cache were A.BCD. And for the longitude, the numbers
she wrote were much bigger, but once again there were four
different pairs of positive integers whose cubes, added in pairs as
above, all added to the same number. The last four digits of that
sum were WXYZ, and my longitude ended with W.XYZ - 10. The Y
digit in my caches coordinates, in other words, was one digit below
Y. I knew then that I'd have to abandon my original plan for this
cache description and substitute Bessie's observation. This was
just too remarkable not to use. That Bessie is one fantastic
mathematician.
The cache contains a log, stash note, pen, geocaching mirror, keys
to Bessie's taxi, a trackable digitalfish token, a trackable niraD
token, and for the kiddies, a pack of baseball trading cards.
Please replace the log in its container carefully, and replace and
cover the cache so it is not visible.