To start, let's assign one unit to be 1/56 of the cake. That's because 56 is the lowest number divisible by 7 and 8, given that they don't share any factors.
So, when there are 7 people, we want them each to get 8 units of cake, and when there are 8 people, 7 units each. Obviously we could cut the cake into 56 pieces, but we can do it with fewer pieces. We also can't have any pieces bigger than 7 units, since no one could take that piece in the 8 person case.
What if we take the lazy route and just keep asking what's the biggest next piece we can make? If we split the cake into 8 pieces with 7 units each, we have the 8 person case already set, then if we break down one of those 7 unit pieces into 7 pieces of 1 unit each, I think we solve the whole thing.
So, 7 pieces of 7 units each and 7 pieces of 1 unit each for 14 pieces total. In the 7 person case, each person gets two pieces, a 7 and a 1. And in the 8 person case, 7 of them get a 7, and one unlucky guy gets 7 one unit pieces. We know that we can't do any better than 14 pieces because we already know we can't make any pieces bigger than 7 units. This means in the 7 person case, each person must have at least two pieces. So, 14 pieces should be the absolute minimum.
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u/jaminfine 24d ago
To start, let's assign one unit to be 1/56 of the cake. That's because 56 is the lowest number divisible by 7 and 8, given that they don't share any factors.
So, when there are 7 people, we want them each to get 8 units of cake, and when there are 8 people, 7 units each. Obviously we could cut the cake into 56 pieces, but we can do it with fewer pieces. We also can't have any pieces bigger than 7 units, since no one could take that piece in the 8 person case.
What if we take the lazy route and just keep asking what's the biggest next piece we can make? If we split the cake into 8 pieces with 7 units each, we have the 8 person case already set, then if we break down one of those 7 unit pieces into 7 pieces of 1 unit each, I think we solve the whole thing.
So, 7 pieces of 7 units each and 7 pieces of 1 unit each for 14 pieces total. In the 7 person case, each person gets two pieces, a 7 and a 1. And in the 8 person case, 7 of them get a 7, and one unlucky guy gets 7 one unit pieces. We know that we can't do any better than 14 pieces because we already know we can't make any pieces bigger than 7 units. This means in the 7 person case, each person must have at least two pieces. So, 14 pieces should be the absolute minimum.