Math/Art Project: Reproduction and Extension of Tom Verhoeff's Dear Cor Always Busy

Math/Art Project: Reproduction and Extension of Tom Verhoeff's Dear Cor Always Busy

This is a collaborative post written by Izhar, Mahnaz, and Malik.

The artwork we chose

We chose Tom Verhoeff's Dear Cor Always Busy from the Bridges 2026 Math Art Gallery. The piece arranges the four words Dear, Cor, Always, Busy in all 

4!=24

4!=24 possible orders, each word given its own colour. It looks like a poem, but every row is determined by a systematic rule.

Our reproduction

We reproduced the artwork by building it from scratch in LaTeX. We used the same four words, the same four colours (Dear = green, Cor = yellow, Always = blue, Busy = pink), and the same alphabetical ordering of the 24 rows. Reproducing the piece by hand taught us how easy it is to repeat a row or skip one, which is exactly why a systematic method is needed.

Our extension

We extended the artwork in two directions, asking two questions the original does not ask:

Parity. If we count the number of inversions in each arrangement (pairs of letters that are out of alphabetical order), we can colour each row by whether its inversion count is even or odd. The result is a perfect 12/12 split , a visual checkerboard with surprising depth.

Necklaces. If we imagine the four words written around a circle, then many of the 24 linear arrangements are the same circular arrangement read from a different starting point. Grouping the 24 rows into these families gives exactly 6 necklaces, each containing 4 rows. We rendered this as a circular mandala in which each colour joins the 4 rotations of one necklace.

The reproduction and the two extensions now live together as a diptych: the linear list on one side, the circular mandala on the other, and the parity colouring as a parallel lens.

Class Activity: Completing the Missing Rows in Traversing the Even’s Algorithm

The class activity asks our classmates to step inside the artwork rather than admire it from the outside. By leaving rows 5–8 and row 13 blank between completed rows, we turn the table into a series of checkpoints: each group can test their reasoning and understanding against the rows we have given. We chose not to walk through the missing rows in advance, because we want the pattern to emerge through discussion. The moment a group notices Dear's back-and-forth sweep is, we think, the moment the algorithm stops being a list of rules and becomes a structure they can predict. Row 13 is the real test of understanding, since it follows a less obvious turn in the sequence. We expect some productive disagreement within groups, and we are curious whether the colour-coded words, like the ribbons in Verhoeff's piece, make the underlying mathematics easier to see. Where groups stall, what they argue about and what they notice first will tell us how well visual art can make an abstract algorithm tangible.

Snags and observations

The first time we tried listing permutations by hand, we repeated and missed rows. This is exactly why the systematic ordering matters, a useful lesson we will bring into our own teaching.

We initially wanted to extend to five words (120 arrangements), but that made the artwork unreadable. The four-word version is denser mathematically and easier to see.

We discovered that both of our extensions are the same 24 objects viewed through different lenses, one linear-arithmetic (inversions), one geometric-rotational (necklaces). That realization shaped the whole presentation.

Colouring by parity makes the checkerboard structure visible but not trivial. The eye cannot simply alternate; the pattern is scattered. This was the most visually interesting part of the whole project.