Deepavali Kolam Maths: 113 Dots, Four-Way Symmetry and One Unbroken Line

Deepavali falls on Sunday, 8 November 2026, and in the week before it, kolam start appearing on porches and in shopping mall lobbies across Malaysia. Made with rice flour, rice grains or semolina and coloured powders, a kolam is there to welcome guests and, as Kuala Lumpur kolam artist Ruben Prakash told The Star in 2024, especially the Goddess Lakshmi. The flour also feeds small insects and birds, which is a generous thing for a decoration to do.

It is also a surprisingly serious piece of mathematics. Not every kolam is drawn the same way, and many of the big festive ones are filled-in blocks of colour. But the older line-and-dot style, known as pulli or sikku kolam, follows strict rules: lay out a grid of dots, then draw a line that loops around every dot and closes up with no loose ends. Rules that tight are exactly what mathematicians like to play with, so let’s count some dots.

A classic layout is a diamond. The rows have 1 dot, then 3, then 5, growing by two until the widest row, then shrinking back down. How many dots is that? With a widest row of 7, the rows are 1, 3, 5, 7, 5, 3, 1, and you can add them in your head to get 25. You don’t need to add them at all, though, because of an old pattern: odd numbers starting from 1 always add up to a square. 1 + 3 = 4, 1 + 3 + 5 = 9, 1 + 3 + 5 + 7 = 16.

So split the diamond across its widest row. The top half, widest row included, is 1 + 3 + 5 + 7 = 16, which is 4 squared. The bottom half is 1 + 3 + 5 = 9, which is 3 squared. Together that’s 16 + 9 = 25. Every diamond works the same way, as two neighbouring squares added together. A widest row of 9 gives 25 + 16 = 41 dots. A widest row of 15 gives 64 + 49 = 113 dots. (That first example also happens to be 3 squared + 4 squared = 5 squared, the same numbers as the famous 3-4-5 right-angled triangle.)

113 dots sounds like a lot to keep track of, crouched on a porch with a handful of flour. Symmetry is what makes it manageable. A diamond has one dot in the exact centre, and if you turn the whole pattern a quarter of a turn, every other dot lands on another dot. So those other 112 dots come in four matching groups of 28. Design one quarter, repeat it three more times, and you have made only about a quarter of the decisions. That’s the same habit behind good congak: don’t do four sums if one sum and a multiplication will do.

Now for the line. Drawing large kolam without lifting your hand was once a matter of pride. Can you tell, before you start, whether that is even possible? You can, thanks to a puzzle from 18th-century Europe. In 1736 the Swiss mathematician Leonhard Euler looked at the seven bridges of Königsberg and asked whether anyone could walk across each bridge exactly once. He turned the map into points and lines, then counted how many lines met at each point. The Königsberg counts were 3, 3, 3 and 5, and that settled it: no such walk exists.

His reasoning fits in one sentence. Every time your line passes through a point it uses up two ends, one going in and one coming out, so a point with an odd number of ends can only be where the line starts or finishes. That means a figure can be drawn in one stroke only if it’s all joined up and has either no odd points or exactly two. Königsberg had four, so it was doomed. Many people count this paper as the start of graph theory, the same branch of maths your navigation app runs on.

Look at a sikku kolam with that in mind. Wherever the line crosses itself, exactly four ends meet, and four is even. Every point in the pattern is even, so nothing rules out a single stroke that finishes where it began. The one thing that can stop it is the first condition: a pattern can fall apart into two or more separate loops, and then you have to lift your hand. Which dot layouts and crossings give one loop and which give several is the hard part, and it is real research. In a 2023 paper, Venkatraman Gopalan sorted square-tile sikku kolam by their symmetry and derived formulas for counting how many there are of each type.

Computer scientists got here even earlier. In 1974, Gift Siromoney, Rani Siromoney and Kamala Krithivasan published a paper called “Array Grammars and Kolam”, which treated these designs as a kind of language: a small set of rules that can generate a whole family of pictures. It’s a neat thought that patterns handed down through families at the doorstep ended up in a computer graphics journal.

If you get a quiet moment over the Deepavali break, try it with a pencil. Mark out the 13-dot diamond (rows of 1, 3, 5, 3, 1), then see if you can loop around every dot in a single closed line. And if you like the feel of dots and lines, Dots & Boxes at cerdikcerdik.com/games/dots-and-boxes is a good place to continue, with a bit of daily congak at cerdikcerdik.com/math to keep those sums quick. Happy Deepavali to everyone celebrating.