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Who found three of four caches?


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Let's say that we want to recognize geocachers who visit a certain number of a set of caches without requiring that they visit them all. For instance at the local club's annual picnic, you many want each of several teams to visit three of a set of four caches.

One way of recognizing those who have done so is to use what's known as "secret sharing". In the case of three of four caches, require each team to come up with all six of a list of six words, and placee the words (here denoted as "word A" through "word F") in the caches as follows:

Cache alpha: Word A, word B, word C

Cache beta: Word A, word D, word E

Cache gamma: Word B, word D, word F

Cache delta: Word C, word E, word F

 

Any team that reaches one cache will have three of the six words. Any team that reaches two caches, regardless of which two they are, retrieves five of the six words. Only a team that visits (any) three of the four caches finds all six of the words.

In practice you'd probably want to use a different set of six words for each team.

 

This is fairly easy to generalize to require visiting n-1 of n caches. The principle is that any team that has visited at most n-2 of n caches must have left 2 caches unvisited. We make a list of n*(n-1)/2 words, and place the words in the caches so that for any pair of caches there is a word that is contained in those two caches and nowhere else. Any team that leaves some pair of two caches unvisited will thus not have found the words that is contained only in those two caches.

To require visiting four of five caches, we need 5*4/2=10 different words. 10 different pairs of caches can be formed from five caches and we assign one of the 10 words to each pair.

(alpha,beta): word A [contained in Cache alpha, Cache beta and nowhere else]

(alpha,gamma): word B

(alpha,delta): word C

(alpha,epsilon): word D

(beta,gamma): word E

(beta,delta): word F

(beta,epsilon): word G

(gamma,delta): word H

(gamma,epsilon): word I

(delta,epsilon): word J

so the contents of the five caches are:

Cache alpha: word A, word B, word C, word D

Cache beta: word A, word E, word F, word G

Cache gamma: word B, word E, word H, word I

Cache delta: word C, word F, word H, word J

Cache epsilon: word D, word G, word I, word J

 

--mammal

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