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Digits of Pi Mystery Cache

Hidden : 9/6/2006
Difficulty:
5 out of 5
Terrain:
1.5 out of 5

Size: Size:   micro (micro)

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Geocache Description:

THE PUBLISHED LOCATION IS NOT THE REAL LOCATION. The real location is within one aerial mile of the published location, at a spot defined mathematically below. There is a subtle hint regarding the cache site in the final paragraph.

The spot is outside and publicly accessible. Nearby parking spaces are available after 5:00 pm and on weekends. GPS reception is sometimes poor at this spot, so you may have to extrapolate from 100 feet away. At the cache site, please do not walk on unpaved areas. The micro-sized cache is contained in a small-sized enclosure that fits naturally into this setting.

The actual coordinates of the geocache are: N 37.X degrees (i.e., 37 + X degrees) W 122.Y degrees (i.e., 122 + Y degrees) where X and Y are fractions between 0 and 1 defined as follows: Let M = 8,468,347, and let N = 8,399,285. Consider the mathematical constant Pi expressed in hexadecimal (base-16) digits instead of the usual decimal (base-10) digits, i.e. Pi = 3.243F6A8885A308D3... Then X is the fraction after the "decimal" point has been shifted to the right by M places, and Y is the fraction after the "decimal" point has been shifted to the right by N places. For example, if M were 4, then shifting the "decimal" point by four places in Pi = 3.243F6A8885A308D3... gives the hexadecimal fraction .6A8885A308D3..., which after conversion to decimal notation yields .4161456606896... A tool to convert hexadecimal fractions to decimal fractions is available here: Conversion tool.

Mathematically speaking, let frac(x) denote the fractional part of x -- for example, frac(3.14159) = 0.14159. Then X = frac (16^M * Pi), and Y = frac (16^N * Pi).

It is important to note that it is not necessary to compute Pi to millions of digits accuracy, in either base-16 or base-10 notation. Instead, the fractions (or base-16 digits) required above can be computed directly and rapidly by using the "BBP algorithm for Pi". The BBP algorithm for Pi is based on the "BBP formula for Pi", which was first discovered by Simon Plouffe in 1995. Both were described in a paper published by Bailey, Borwein and Plouffe in 1996.

There are several descriptions of the BBP formula and the BBP algorithm on the Internet. See, for example: BBP paper. You are invited to try writing your own computer program implementing the BBP algorithm. Alternatively, you can utilize one of the computer implementations available on the Internet, such as those available at BBP code site. You need to compute X and Y to at least five digits accuracy to locate the cache. As a check that your program is working correctly, the first two digits of X and Y (in decimal notation) are: X = 0.88..., Y = 0.24... You can also check your answers for this puzzle here: Geochecker site.

It is worth noting that the BBP formula for Pi was itself discovered using a computer program -- this is likely the first instance where a significant new formula for Pi was discovered by computer. This computer program was based on the "PSLQ" algorithm of mathematician-sculptor Helaman Ferguson. PSLQ is now regarded as one of the most important algorithms to emerge from 20th century mathematics, and is widely used in the emerging field of "experimental mathematics". Ferguson’s sculptures may be viewed at Ferguson sculptures.

Additional Hints (Decrypt)

Fznyy snxr ebpx, nccebk rvtug srrg sebz gur fphycgher, ntnvafg jnyy, haqre cyndhr

Decryption Key

A|B|C|D|E|F|G|H|I|J|K|L|M
-------------------------
N|O|P|Q|R|S|T|U|V|W|X|Y|Z

(letter above equals below, and vice versa)