The plan behind the story · Sixty paper hats on a table in Bridlington: David Smith and the shape mathematicians had sought for decades
Four weeks with the hat: from a first patch to a question of your own
In four weeks you will have built a patch of hat tiles by hand, seen for yourself why it will not settle into a repeating pattern, and started a notebook of shapes. If you already do puzzles or maths for fun, the same four weeks end with a result you worked out yourself and a question sent to someone who can test it.
Why this works
David Smith did not set out to solve a famous problem. He was trying shapes in a piece of software, as he often did, and one of them tiled in a way he had not seen before. Quanta Magazine reported that he then cut 30 copies from card, and 30 more, and fitted them together on a table in Bridlington.
That is the whole method, and it has three parts. Play often enough to know what ordinary looks like. Make the thing with your hands when the screen stops telling you anything. Then ask someone who has a tool you lack: Smith emailed Craig Kaplan, a computer scientist he knew from a tiling mailing list, and the proof came from the group that formed around that email.
This plan asks for no algebra. It asks for scissors and patience.
Before you start
- A screen. Craig Kaplan’s hat page (linked from the story) has a browser app that builds patches of hats and saves them as pictures, and a link to a Mathigon polypad where you move virtual tiles by hand.
- Card and scissors. Cereal-box card is fine. A printer helps. If you have none, a library or a neighbour will do for one sheet.
- A notebook. One page a session.
- A table you can leave covered for a day or two.
- Time. Twenty to forty minutes, three times a week.
The plan
| Week | Do this | How often | Notes |
|---|---|---|---|
| 1 (new to this) | Open the polypad and fit virtual hats together with no aim but leaving no gaps. | 3 sessions of 20 minutes | Write down how many you placed before you got stuck. Getting stuck is normal. |
| 1 (already doing it) | Print a patch from the browser app and cut 60 hats from card, Smith’s number. | 2 sessions of 40 minutes | Colour one face of each tile so you can tell when one has been flipped. |
| 2 (new to this) | Print a patch, cut out 30 hats and cover as much table as you can. | 3 sessions of 20–30 minutes | Flipping a tile over is allowed. The hat needs it. |
| 2 (already doing it) | Build outwards from one tile until it has 3 complete rings round it. | 3 sessions of 40 minutes | Count the flipped tiles and note where they sit. |
| 3 (new to this) | Rebuild your patch from memory, then look for any block of tiles that appears twice. | 3 sessions of 30 minutes | Photograph each attempt. Compare them side by side. |
| 3 (already doing it) | Draw 3 shapes of your own on squared or triangular paper. For each, find how many complete rings of copies will surround it. | 3 sessions of 40 minutes | Kaplan’s 2025 article on arXiv explains the idea (he calls it the Heesch number). |
| 4 (new to this) | Show someone. Give them ten tiles and no instructions. | Once, plus 2 sessions of your own | Explaining it is how you find out what you have noticed. |
| 4 (already doing it) | Write up one question you could not settle, with a drawing, and send it to a named person or group. | Once, plus 2 sessions | A club, a mailing list, a teacher, a researcher. Keep it to one page. |
If it goes wrong
- The tiles will not fit at all. Check the print: every tile must be the same size. Cut inside the line, not outside it.
- You are stuck after eight or nine tiles. Take the last three off and try a flipped one. Everybody backs up.
- You missed a week. Start again at the row you left. The tiles keep.
- Nobody replies to your question. Send it to one more place, and keep working on it yourself in the meantime.
Track it
One page per session: the date, how long you spent, the biggest patch you made (count the tiles), how many were flipped, and one thing you noticed. Keep the photographs with the notes. After four weeks, read the pages in order.
What next
- The second shape. Smith’s follow-up, the spectre, tiles without any flipping. Kaplan’s hat page links to that paper.
- The paper itself. It was published in the journal Combinatorial Theory in June 2024, and the same text is free to read as a preprint. Read the introduction and look at the pictures; skip what you cannot follow.
- Other puzzles. There are four new ones a day on our puzzle page, at three levels.
General information, not medical advice. Check with your GP before starting a new exercise programme.