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# September 2015 - Key Arrangements

There are two men, Mr. Millionaire and Mr. Billionaire. Both have a safe containing their personal valuables. They also have some guards in their mansions. Both want their safe to be guarded with multiple locks and the keys distributed to themselves and their guards.

Mr. Millionaire has four guards and wants a system whereby he can access his safe only if accompanied by any one of his guards and, in his absence, for any three guards to be able to access the safe. However, no two guards should be able to work together and open the safe.

Mr. Billionaire, on the other hand, has even more valuable possessions in his own safe and wants much tighter security. He has seven guards and wants a system whereby he can access his safe only if accompanied by any two guards and, in his absence, for all seven guards to be able to work together and open the safe. However, no six guards should be able to work together and open the safe.

The answer is the product of the minimum number of types of keys that are required in each system.