Use mathematical induction to show that 11 people can

Chapter 5, Problem 53E

(choose chapter or problem)

Use mathematical induction to show that \(n\) people can divide a cake (where each person gets one or more separate pieces of the cake) so that the cake is divided fairly, that is, in the sense that each person thinks he or she got at least \((1 / n)\) th of the cake. [Hint: For the inductive step, take a fair division of the cake among the first \(k\) people, have each person divide their share into what this person thinks are \(k+1\) equal portions, and then have the \((k+1)\) st person select a portion from each of the \(k\) people. When showing this produces a fair division for \(k+1\) people, suppose that person \(k+1\) thinks that person \(i\) got \(p_{i}\) of the cake where \(\sum_{i=1}^{k} p_{i}=1\).]

Equation Transcription:

)

Text Transcription:

n

(1 / n)

k

k+1

(k+1)

k

k+1

k+1

i

p_{i}

sum_{i=1}^{k} p_{i}=1