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The soccer betting form contains a list of 13 games. There are three possible outcomes for each game: “the first team won”, “the second team won” and “draw”. Each betting form allows to chose one outcome for each game.In order to win a big prize, it is required to guess correctly the results of at least 12 games. What is the minimum number of forms Alex has to fill in to ensure that he wins a big prize?

Hint: There exists a perfect Hamming code with parameters [13,10,3] over $\mathbb F_{3}$

Any suggestions on how to solve this puzzle

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Hint: the perfect Hamming code has the property that every vector in $\mathbb F_3^{13}$ is either a codeword in the code or is at Hamming distance $1$ from some codeword. Every possible result of the $13$ games can be expressed as a vector in $\mathbb F_3^{13}$. Now think! – Dilip Sarwate Apr 20 '14 at 14:20
Thanks a lot @DilipSarwate it helps. – kingmakerking Apr 21 '14 at 11:21

More hint.... think that you are planning to win the prize if you have a output-word (of length 13) with error 0 or 1 i.e. at a distance of 1 from the code word. Now you can use the sphere packing bound to find the answer.

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