Three recent articles in the Tips & Techniques series addressed the question of stepwise solutions in Maple. Just what is it that Maple provides by way of stepwise solutions for standard calculations in the mathematical curricula? There are commands, assistants, tutors, and task templates that provide stepwise calculations in precalculus, calculus, linear algebra, and vector calculus. In addition, since Maple can implement nearly any mathematical operation, any stepwise calculation can be reproduced in Maple by assembling the appropriate intermediate steps, just as they would be assembled when working with pencil and paper.

Some calculus texts compute volumes of solids by the method of "slices" before they discuss the methods of disks and shells. On the other hand, there are texts that start with disks and shells, then throw in a few examples of slices. In any event, these calculations are supposed to be illustrations of how definite integration is an additive process. Unfortunately, students often get lost in the details of the individual examples, and fail to see that all these calculations are just demonstrations that definite integration is a process of addition.

Do an internet search on "Challenger Puzzle" and you will find descriptions and solvers for a puzzle that involves sums of integers from one to nine. Indeed, on a 4 × 4 grid where sixteen integers would fit, four are given, along with the row, column, and diagonal sums of the numbers not shown. The object of the puzzle is to discover the missing twelve numbers.

Unlike Sudoku, the digits can repeat. And unlike Sudoku, the puzzle can have multiple solutions. In fact, "There may be more than one solution" is explicitly stated below the directions, copyrighted by King Features Syndicate, Inc., that appear in my local newspaper, the Waterloo Region Record.

Recently, I received an email from a physics instructor asking for help in building a tool that would display the solution of the initial value problem 

with the four parameters under the control of sliders. (Of course, we recognize that this equation governs the damped, driven linear oscillator, and that the request to endow its solution with sliders is in service of visualization of the change in the nature of the solution as the parameters vary.)

It was years since I "derived" the result that slopes of perpendicular lines were negative reciprocals of each other. So I thought it would be easy to show that when , where, in Figure 1, is the slope of line (black) and is the slope of line (red). Clearly, lines and are perpendicular when .