A number of guests gather around a round table for a dinner. Between every adjacent pair...

A number of guests gather around a round table for a dinner. Between every adjacent pair of guests, there is a plate for tips. When everyone has finished eating, each person places half their tip in the plate to their left and half in the plate to their right. Suppose you can only see the amount of tips in each plate after everyone has left. Can you deduce the amount that each individual tipped?

(a) Suppose six guests sit around a table and there are six plates of tips. If we know the amount of tip in each plate, P1 to P6, can we determine each individual’s tip amount, G1 to G6 (G1+G6 = P1, G1+G2 = P2, G2+G3 = P3, ... , G5+G6 = P6)? If yes, explain why by examining the relationship between the plate values, P1 to P6, and guest tips, G1 to G6. If not, give two different assignments of G1 to G6 that will result in the same P1 to P6.

(b) Now lets consider five guests at the table, G1 to G5, and we can see the amount of tips in the five plates, P1 to P5 ((G1+G5 = P1, G1+G2 = P2, G2+G3 = P3, ... , G4+G5 = P5)). In this new setting can you figure out each guests tip values, G1 to G5?

(c) If n is the total number of guests sitting around a table, for which values of n can you figure out everyone’s tip? You do not have to rigorously prove your answer. (Hint: consider what is different about parts a and b.)

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Expert Solution

Colby Messinger answered 1 year ago

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