Mathcad Community Challenge: January 2026: The Art Gallery Problem

The Challenge

The January 2026 Mathcad Community Challenge was inspired by the Louvre heist in Paris:

Using Mathcad Prime, solve any or all of the following challenges:
Challenge 1 (Polygonal): find the minimum number of guards necessary to cover 100% of the following polygonal space.

Challenge 2 (Graphing): using the plotting and charting capabilities of Mathcad, depict the placement of the guards and their zones of coverage.

Challenge 3 (Orthogonal Variant): find the minimum number of guards for the following art gallery.

Challenge 4 (Graphing): Depict the placement of the guards and their zones of coverage for the orthogonal variant.

Challenge 5 (Basher Tarr variant): You are the tech expert for the Ocean’s 11 crew. Your job is to remove one guard to leave the maximum amount of exposed area. Which guard do you remove in either of the scenarios?

Notes on the Solution

The Art Gallery Problem is quite a complicated challenge in computational geometry. The solution for the upper bounds of the number of guards has been known for years. Mathematician Steve Fisk devised an ingenious color-based solution with triangles that can be taught to students.

I should have noted in the comments that I did not expect anyone to solve the problem programmatically. This problem has challenged mathematicians for decades. I thought people could solve the problem visually and then document their results in Mathcad Prime.

For reference, I measured the surface area of the shapes in Creo Parametric. The polygon is 176,400 square units and the orthogonal challenge is 199,800 square units.

The Submissions

As usual, Alan Stevens was the first to submit a worksheet. He uses Mathcad Express. He calculated the upper bound to be 9, but his solution showed the art gallery can be covered by 6 guards. Alan depicted the placement of the guards in an XY Plot with each guard’s zone of responsibility in a different color.

Alan makes his worksheet visually appealing by turning off the grid, changing the font (to my personal favorite Lexend, the font for improved readability), and hiding matrices and functions in collapsed areas.

Although I did not expect anyone to approach the problem programmatically, Professor Tokoro stepped up to the plate with four Mathcad Prime worksheets. I can’t begin to comprehend the solution but it involves programs, matrices, symbolic evaluations, summations, and built-in functions like stack. He depicted his solution as well in an XY Plot. It had 6 guards like Alan, but in different locations, which is to be expected. One thing that makes the Art Gallery Problem so hard is that there are multiple solutions.

As I reflect on often, it is important to make worksheets aesthetically pleasing. Some of the methods that Professor Tokoro uses here include math regions with font color, background color, and borders; and hiding math regions in the Draft View.

6 Ways to Make Your Mathcad Worksheets Publication Ready

Read through Dave Martin’s tips on making your Mathcad worksheets ready to show to others.

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The Basher Tarr variant for this arrangement would give the Ocean’s 11 crew less than 3.1% of uncovered ground.

Professor Tokoro’s second worksheet reflected additional time to think. There are additional programs for painting the portions of the XY Plot. The math regions are color-coded with the different assigned guards. The worksheet and plot are very pretty; I like the choice of pastels.

The orthogonal challenge showed 7 guards could cover the entire area. The Basher Tarr variant would leave a maximum of only 3.5% of the floor uncovered. This suggests an interesting aspect of the Art Gallery Problem to me: even with a minimum number of guards, there must be significant overlap between sectors of view.

The third worksheet from Professor Tokoro added color coding for the orthogonal challenge. The worksheet is so elegant that it hides how complicated the functions and the plotting are. The fourth and final worksheet had some changes, but I could not tell what they were.

An aspiring person might want to take Professor Tokoro’s paint programs and put them into a library to use in other applications. If you aren’t aware, Mathcad Prime provides commands like Include Worksheet to leverage user-developed functions and programs in new worksheets.

What We Can Learn

For computational geometry problems that may appear too hard to be solved programmatically, Mathcad Prime can be used to visualize and communicate the results. The Art Gallery Problem can be solved by humans with visualization and color coding. Ostensibly, it would seem that this problem could not be solved programmatically in Mathcad Prime. As it turns out, Mathcad Prime does have the functionality to solve these kinds of problems.

For anyone who wants to increase their ability to solve problems in Mathcad Prime, I strongly recommend they learn keywords for symbolic evaluation, programs, and functions for manipulating matrices.

Join us in March for another challenge, and be sure to learn from all of the previous challenges we’ve had so far!