Splitting 25 Into Pieces that Maximize the Product
Based on the problem: find a set of numbers that add up to 25 and have a maximal product out of all such sets. The applet below shows how dividing 25 (or any number A you chose) into n pieces changes the product. The product of dividing A into n pieces is shown on the y axis, and the number of pieces n is shown on the x axis. Do you notice how for optimal n, A/n is close to e? Are there values of A where there are two or more solutions? Note: in case you're wondering, the code for "Auto Zoom" is On Click: ZoomIn[-.1*A,-g(A/e)*.1,A*1.2,g(A/e)*1.2].