# The cake is a lie!

Suppose a group of $$n$$ people want to split a group of whole cakes among each other. Each cake is of one type only e.g. vanilla, chocolate, strawberry, etc.

Each person also has a value function associated with each type of cake, which denotes how much each cake is "worth" to each player. The sum of all the value functions for a given person is $$1$$, and every person values nothing at $$0$$. Note that a value function must be $$\geq 0$$ and $$\leq 1$$.

For example, person 1 might value vanilla at $$0.05$$, while person 2 values vanilla at $$0.8$$. Person 1 might value chocolate at $$0.77$$, while person 2 values chocolate at $$0.00002$$.

In addition, a person values a fraction $$f$$ of a cake at a fraction $$f$$ of the value function he/she associates it with. For example, given the above value functions, person 2 would value half a vanilla cake at $$0.4$$, while person 1 values $$\frac{1}{7}$$ of a chocolate cake at $$0.11$$.

The total value a person receives is the combined values of all the cakes he/she has, based on his/her own value functions. For example, if person 1 receives a vanilla cake and a chocolate cake, his total value would be $$0.05 + 0.77 = 0.82$$. If person 2 receives half a chocolate cake and half a vanilla cake, her total value would be $$0.00001 + 0.4 = 0.40001$$.

Is it possible to divide the cakes among all $$n$$ people such that each person receives at least a total value of $$\frac{1}{n}$$?

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