Problem of the Week
Problem B
Equal Cake and Icing

Serenity is having \(16\) guests for dinner. She baked a cake for dessert using a square cake pan with side length \(36\text{ cm}\). The cake is \(8\text{ cm}\) tall. The top face and side faces of the cake are covered in icing.

She would like to slice the cake into \(16\) pieces. She calls a slicing a "fair cake" if each piece has the same amount (volume) of cake and the same amount (surface area) of icing.

1. To cut the cake into \(16\) pieces, suppose she makes three equally-spaced vertical slices and three equally-spaced horizontal slices through the top face of the cake. Is this a fair cake?

   

2. To cut the cake into \(16\) pieces, suppose she first divides each edge of the top face into four equal lengths. She then makes a straight slice from each end of a length, through the centre of the square, to an end of a length on the opposite edge. Is this a fair cake? Show calculations to support your answer.

   

Extension:

Only \(9\) guests want to eat dessert. Serenity decides to cut the cake into \(9\) pieces by dividing the entire perimeter of the cake into nine equal lengths, starting in the top-left corner and moving clockwise. She then makes a slice from each end of a length to the centre of the square. Is this a fair cake? Show calculations to support your answer.

Theme: Geometry & Measurement