Sum of Cubes to Square of Sum

thingiverse

####The Sum of Cubes to Square of the Sum The sum of a sequence of cubes such as *1^3 +2^3 +3^3+4^3* is the square of the sum *(1+2+3+4)^2*. In other words, the sum of the first n cubes is the n-th triangular number squared. We can prove it using a bit of algebraic manipulation. Nonetheless, a physical model is equally interesting! Included in the present design are 12 pieces that can be arranged into a 10 by 10 square or four cubes (1-cube, 2-cube, 3-cube, and 4-cube). Each unit cube is 10mm^3. It works like a simple puzzle for young children, who can learn a host of ideas in number sense and perhaps more. ####Challenge There are certainly many ways to dissect a 10-by-10 square and rearrange the pieces into 4 cubes. Now, what is the **least number of pieces** one can dissect a 10-by-10 square to do that? For a good-looking equation, here is the LaTeX code: \begin{align\*} 1^3 + 2^3 + \cdots +n^3 &= (1+2+\cdots+n)^2\\ &=\left(\frac{n(n+1)}{2}\right)^2 \end{align\*} Have fun!