Coin touches coins

What is the fewest number of coins that would be required in order to make sure each and every coin touches exactly three other coins?




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A number with an interesting property:

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When I divide it by 3, the remainder is 2.
When I divide it by 4, the remainder is 3.
When I divide it by 5, the remainder is 4.
When I divide it by 6, the remainder is 5.
When I divide it by 7, the remainder is 6.
When I divide it by 8, the remainder is 7.
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When I divide it by 10, the remainder is 9.

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There is a cryptic organization called Cicada 3301 that posts challenging puzzles online, possibly to recruit codebreakers and linguists.