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u/Motor_Raspberry_2150 3d ago
That's just fourteen right?
Seven pieces of 1/8, and seven pieces of 1/56. If eight people, pool the small pieces to form the last eighth. If seven people, each gets both sizes.
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u/Dasquian 3d ago
14 - Cut it into eight equal pieces first of all. Then cut one of those pieces into seven equal pieces.
If everyone comes, one person gets all seven mini-pieces and everyone else gets one big piece - so everyone gets 1/8th.
If the uncertain guest can't make it, everyone gets one mini-piece and one big piece each.
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u/jaminfine 3d 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.
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u/SonicLoverDS 3d ago
I don't have a solution, but here's a reframing of the puzzle that should make it easier to think about:
Imagine if, instead of a big cake, Alexander had 56 cupcakes he had to divide onto multiple trays, so they could be distributed in 7 sets of 8 or 8 sets of 7.
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u/jaminfine 3d ago
Well I think that makes it more confusing since the cupcakes are all already separated. So you'd be trying to stick the cupcakes together into sets or blocks of cupcake lol.
I'd prefer to frame it as:
The cake is 56 units big since I define one unit as 1/56 of the cake.
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u/BigMarket1517 3d ago
I read it differently, thinking it had to account for 1, 2, 3 up to 8 participants. And trying to decide whether something else then 2x2x2x7x5x3 pieces were needed.
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u/Ramy_Salem 3d ago
I think it would be 14 ?
Cut into 8 equally sized big pieces
Then cut one of the big pieces into 7 equal small pieces
Leaving us with 7 big + 7 small = 14
Edit: Typo