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Grandi's Series Mystery Cache

Hidden : 12/14/2024
Difficulty:
3 out of 5
Terrain:
1.5 out of 5

Size: Size:   micro (micro)

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Geocache Description:


Grandi's series

In mathematics, the infinite series 1 − 1 + 1 − 1 + ⋯, is sometimes called Grandi's series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703. It is a divergent series, meaning that the sequence of partial sums of the series does not converge.

However, though it is divergent, it can be manipulated to yield a number of mathematically interesting results. 

One obvious method to find the sum of the series

1−1+1−1+1−1+1−1+…

would be to treat it like a telescoping series and perform the subtractions in place:

(1−1)+(1−1)+(1−1)+(1−1)+…=0+0+0+0+…=0.

On the other hand, a similar bracketing procedure leads to the apparently contradictory result

1+(−1+1)+(−1+1)+(−1+1)+…=1+0+0+0+…=1.

Thus, by applying parentheses to Grandi's series in different ways, one can obtain either 0 or 1 as a "value".

Remember (not everything) or (everything) can be black or white.

Sits and Thinks

Turning over the parchment yields the following code:

The Code

How to solve:
1) Transcribe the image with all the numbers :-) 
2) It's a simple binary puzzle.  Every number either "exists" or it "doesn't exist"
3) When you have what "exists", stand back and admire your handiwork for the ah-ha moment.

Additional Hints (Decrypt)

Chmmyr: Q0a'g 0i3egu1ax 1g. 1g'f a0g n5 te4aq1 n5 l0h z1tug gu1ax. Vg'f 2 y4l3e3q Pnpur: FCBE

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)