Srinivasa Ramanujan (1887-1920) was the most famous self-taught mathematician of the last century. After struggling in his native India he submitted hundreds of his mathematical results to the famous pure mathematician G. H. Hardy who, though skeptical, invited Ramanujan to join him in England.
The story goes that after taking a cab Hardy remarked to Ramanujan that the cab number, #1729, was rather uninteresting. After a brief pause Ramanujan replied "No. It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways" ( 1729 = 1^3+12^3 = 9^3+10^3 ) Hardy was convinced. Ramanujan went on to prove over 3,000 mathematical theorems during his stay in England.
Since that time, mathematical/recreational enthusiasts, and more recently geocachers, have enjoyed taking an arbitrary number and trying to find out something interesting about it. To that end, we first found a central location to place this cache and then tried to find what was interesting about the numbers in its coordinates. Here are the results.
a = 333 in base 5.
[To convert a number abc from base 5 to decimal, evaluate a X 5^2 + b X 5 + c. ]
b = 3 X 3 (three times three; too easy?)
c = two gross!
d = the number of derangements of ABCDE.
[On derangements: Suppose five geocachers named A, B, C, D and E brought a coin apiece to a coin swap wishing to swap coins in a way so that nobody got his/her own coin back. Suppose A got B's; B got D's; C got E's; D got C's and E got A's. Call this exchange BDECA. Got it? Then the question becomes: in how many different ways can A,B,C,D,E be re-arranged so that none of the letters is in its original position? With only two cachers, d = 1 for the only exchange is BA. With three cachers, d = 2 for BCA and CAB. With four, we could have BADC, BCDA, BDAC, CADB, CDAB, CDBA, DABC, DCAB and DCBA, so d = 9. For five, d = ?]
e = 3^3 (i.e. 3 X 3 X 3)
f = the smallest number expressible as the sum of the squares of two distinct primes in two different ways.
[Huh? The first few primes (a prime is any integer > 1 which is evenly divisible only by itself and by 1) are 2, 3, 5, 7, 11, 13, 17, 19, 23, ... . We're looking for f, a 3-digit number, for which f = p^2 + q^2 for two different primes p and q.]
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