Question 1206044: Mr. Smith had just finished baking a cake for the mathematicians’ banquet. The cake was specially designed in the shape of a cube. In the process of carrying the cake to the frosting table, Mr. Smith suddenly slipped and the cube-cake went sailing into the vat of chocolate frosting. Mr. Smith thought quickly and then yelled “FIRE!” (He knew that no one would come to help if he yelled, “Chocolate”).
Almost immediately help arrived and the cake was fished out of the chocolate. Fortunately, the cube-cake was still in one piece, but was now frosted on all sides.
Mr. Smith proceeded to the banquet hall with his unusually frosted cake in hand.
The mathematicians were delighted when they saw the cube-cake with all of the frosting. They asked Mr. Smith to stay and cut the cake.
One of the mathematicians, Mrs. Hayne suggested that the cake be cut into cube-shaped pieces, all pieces the same size. Mr. Smith agreed, but before cutting the cake he turned to Mrs. Hayne and asked how many mathematicians would like a piece without frosting, a piece with only one side frosted, a piece with exactly two sides frosted, or a piece with three sides frosted.
Being a mathematician, Mrs. Hayne responded, “Cut the cake so that the number of pieces without frosting is equal to eight times the number of pieces that have frosting on three sides. You will then have enough of each type of piece to satisfy everyone, with nothing left over.”
Please answer the following questions.
How many mathematicians attended the banquet?
How many mathematicians were served a piece of cake without frosting? How many mathematicians requested a piece with only one side frosted? How many were served a piece of cake with exactly two sides frosted? How many requested a piece with three sides frosted?
Answer by math_tutor2020(3816)   (Show Source):
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