Question

In a high school science activity involving 100 students, each student is given a piece of plasticine in the shape of a cube so that the nth student has a piece with edge length n cm. This means the 1st student ha s the piece with edge length 1cm, the 2nd student has a piece with edge length 2 cm and so on. The teacher asks the students to combine all pieces to make one large cuboid-shaped model with edge length an integer number of centimetres. Can they succeed?

Answer 1

Well, by direct computation, the total volume of the boxes is \[1^3+2^3+3^3+...+100^3\]This is equal to the expression \[\left({100(101) \over 2}\right)^2 =25502500\]This is not a perfect cube, so they can not succeed.