Combinatorics problem - Distribute cakes amongst peers I'm having a mind wrenching question that I just cannot answer. It's been a while since I was at the school bench so I wonder if anyone can help me out? :)
We have 10 students with 5 cakes each to be shared amongst each other.
The students can give the cakes out, but they can’t give a piece to a person who gives them a piece (and vice versa). They also can't give more than one piece for the same person. 
How can they distribute the cakes so that everyone gets as many cakes as possible? 
 A: Place the kids in a circle, numbered 1 to 10. Have the odd numbered kids pass four cakes, one to each of the four closest students on their right (so 1 passes to 2,3,4,5). Have the even numbered students pass five cakes to the five students on their left (so 2 passes to 3,4,5,6,7).
The odd students receive 5 cakes, and the evens receive 4. This is optimal; there are 45 pairs of students, so only 45 cakes can be passed, meaning only 4.5 cakes can be received on average.
A: The first solution that comes to mind is for no one to give any cakes to anyone, so everyone "gets" (to keep) the five cakes they brought. But I assume you want "gets" to mean "gets from someone else."
If this is the idea, no one can give out all five cakes and get five back, because that would require there to be 11 students. If everyone must get the same number of cakes, the highest that number can be is four.
Sit the students at a round table and have everyone give one cake to each of the four people to their left. Each student will then receive four cakes, one from each of the four people on their right.
