Courier-ing favor
Mike Pence has an agent in Russia he is using to undermine and overthrow the evil Putin dictatorship (once Putin is gone, Pence will stage a bloodless 25th Amendment coup and replace Trump as President.) Pence has a new secret code that comes in 5 parts (A, B, C, D and E) to communicate with his agent. He wishes to courier the parts in from the US to his Russian agent. All 5 parts have to reach the agent for the code to be useful.
Putin's thugs sometimes intercept the couriers. If a courier is captured, Putin gets all the parts to the code the courier is carrying. If Putin captures all 5 parts, it will be disastrous.
Assume that no more than 2 couriers will get captured. What is the minimum number of couriers Pence must use to successfully get all 5 parts of the code to his agent while ensuring that Putin does not get all 5 parts for himself?
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1 solution
Since 2 couriers could be captured, each part must be carried by at least 3 couriers to ensure that one courier carrying the part reaches the agent.
We show that 7 couriers is feasible:
Courier 1 carries A, B, C
Courier 2 carries A, D
Courier 3 carries B, D
Courier 4 carries C, D
Courier 5 carries A, E
Courier 6 carries B, E
Courier 7 carries C, E
To show that 7 is the minimum: Since there are 15 parts needed (3 copies each of A, B, C, D, E = 3 * 5 = 15 parts), any solution with 7 or fewer couriers will have at least one courier carrying 3 parts. WLOG assume those parts are A, B, C.
3 couriers must carry a copy of D. And 3 couriers must carry a copy of E. But no courier can carry both D and E, because otherwise if that courier and the one carrying A, B, C are both captured Putin would get all five parts to the code. So the minimum number needed is 1 + 3 + 3 = 7 couriers.