Colouration

Preface

Introduction

Terms and Definitions

Artificial Intelligence

Part 1 Abstract Colouring
Map Colouring With 3–8 Colours
(i) Three Colours
(ii) Four Colours
(iii) Five Colours
(iv) Six Colours
(v) Seven Colours
(vi) Eight Colours
Polycairo Colouring
(i) Dicairos
(ii) Tricairos
(iii) Tetracairos

Combining (Hybrid)

Various Published Coloured Diagrams in Books and Articles

Part 2 In Situ Colourings

Acknowledgements

Bibliography

“Whom the gods would destroy, they first make mad"

William Anderson Scott, Daniel, A Model For Young Men, 1854, although it has roots in classical Greek tragedy

Preface
As an aside, the aphorism is most apt. For the most part, this study — with its apparent simplicity — is like nailing jelly. An envisaged “day or so” turns into days, then weeks, and months and still it refuses to resolve. It drives me mad. It is yet another of those studies that expand far beyond what was intended: do I abandon days of work when, with one more push, it might finally yield? And then, when it doesn’t, the cycle repeats. Even a nominal cessation leaves matters unfinished, which pains me no end. It is frustration personified, all from what is, at face value, merely colouring a simple tiling. Although perhaps it is more difficult than it appears. Late in the day, I prompted AI about the perceived difficulties, which remarked that “tilings with mixed vertex types are almost always harder to colour systematically”. I shall take that in my defence. With previous colouring studies — the Pólya bird tile fiasco of 2006, which took a year — I should have known better. I never learn. Every study now seems to turn into an epic, despite my best intentions of relative brevity.

Even the simplest three‑colour case presents problems in systematic determination, with vast combinatorial sets. This particular instance is best approached by “smart thinking” rather than a brute‑force enumeration by hand of all possibilities. Aside from the basic three‑colour result, the four‑colourings remain unresolved, though considerably advanced. I have likely found most, but I am far from certain. It is an unsatisfactory state of affairs, yet to continue to the point of absurdity in pursuit of proof is madness, given the relative unimportance of the matter. It is a problem better suited to a computer, but well beyond me to set up. And all this without delving into higher colour counts — five, six, and beyond. I simply want to tidy up and leave the study in a state that is understandable for a later return, without having to unravel my intentions afresh after forgetting the intricacies.

The previous incarnation (2013, archived at the Wayback Machine) serves to a degree, but is far from ideal: basic, with minimal explanatory text. A three‑colouring was shown without proof, three four‑colourings, two five‑colourings, and one six‑colouring. This new text serves as raw material, neatly written‑up and ready to go, albeit with judgement required as to what to include. Further complications arise with the “optimum transfer” of new diagrams to the webpage. There are so many combinations: GIF, PNG, or JPG — and at what pixel dimensions? An initial upload produced fuzzy images, requiring a redo. A simple upload appears tiny. I then have to “unlock” the image and drag it manually into position, rather than simply insert it. Inevitably, there are minor differences in image size, but without knowing that in advance, one would never discern it. It is far from ideal. And just when you think the matter is under control, uploading throws another curveball. The problems, mathematical and presentation, are never‑ending.

Introduction
Of mathematical interest are the colouring possibilities of the Cairo tiling—map‑colourings (3–8 colours inclusive), polycairos, including the “Macmillan” colouring and, more recently, combinations of these, which I title as “hybrids”. My primary concern here, however, is with the map‑coloured instances.

The literature on this topic is surprisingly extremely scant. In most discussions the tiling is shown in wireframe mode, and when coloured at all, it is typically presented as a single arbitrary four‑colour example, with all tiles in the same orientation—this being the easiest to imagine, as no real thought is required. So far as I am aware, only Rinus Roelofs (1998) and George Baloglou (2013) have examined these colourings in a considered way, though to different extents: Baloglou’s work is by far the more detailed, albeit solely focused on six‑colourings, while Roelofs treats the matter essentially in passing, with a brief feature on four‑colouring among other discussions in a short, two-page article. Neither study, however, can be described as “all‑encompassing” in that all possibilities for a given 4-, 5-, 6-colouring… are not stated. For the basics, the colourings are not developed from first principles—specifically, from the foundational cases of three and four colours. I would be indebted to any reader who can refer me to further work in this field, or who may be interested in studying this aspect themselves.

In contrast to the above, I begin here from first principles, with the three‑ and four‑colour cases in relative depth, before paying lip service to the higher orders (which are judged impractical to complete without computer assistance, i.e. to determine the admissible colourings). It will be seen that three colours admit only a single instance, whereas four colours yield at least six distinct possibilities (which incidentally doubles the known number of colourings). Whether this enumeration is all is unclear. Proving this is another matter entirely. Although 4‑colouring, with four orientations of the tile, may appear at first sight to be a straightforward problem, it is anything but. Complications arise from the tiling’s mixture of three‑ and four‑valent vertices. For colouring purposes, the tiling is perhaps best considered initially in its simplest abstract form—a double basketweave (Fig. 1)—before transposition to the Cairo tiling itself, which is the approach I adopt. For the pentagonal tile, I have used the in situ model rather than other models, such as the equilateral or the dual of the 3.3.4.3.4 tiling, although these could be used.

In the text, I outline two methods, simply stated as “smart” and “dumb”. Smart refers to an approach that uncovers valid colourings without the drudgery of checking every possibility by hand—a process which, even for the lower datasets, quickly runs into thousands of configurations. Dumb refers to the brute‑force alternative: examining all datasets exhaustively. This guarantees completeness, but at a prohibitive cost in time and effort. One might argue for calling the latter “Dumb‑Smart”, but for clarity I retain the original terminology.

The clearest illustration is the 3‑colour case. The dumb approach does indeed find all possibilities, but only after many days of labour. The smart approach resolves the matter in minutes. However, for higher colourings—4, 5, 6, and beyond—I have not found a viable smart method, and so the time‑consuming dumb approach becomes necessary. Even then, for the 4‑colourings I was unable to complete a full enumeration. For higher orders still, the number of possibilities rises exponentially to the point of practical impossibility.

As noted earlier, this is fundamentally a problem suited to computer enumeration rather than human persistence.

Preliminaries

For the study, I use various stock diagrams, terms and definitions as set out below.

The Two Core Diagrams
The default presentation is of the Cairo tiling and its double basketweave equivalent. This is generally used for showing complete colourings.

Wireframe Cairo tiling and Basketweave equivalent 

“Ribbon” Arrangements

When discussing the colourings, a common feature I describe as “ribbon colouring” can be seen below. An alternative could be “threads”. Whatever, the premise is made clear below: two types of ribbon are discerned, infinite (in principle) and partial, consisting of a smaller unit, with a 2-unit shown below. 

Artificial Intelligence
A seemingly obvious time‑saving idea is to use AI to solve this problem, with prompts along the lines of “Colour this basketweave tiling with four different colours so that no tile of the same colour is adjacent to another.” However, the readily available models — Copilot, Gemini, Grok, and ChatGPT — are, in this respect, a complete embarrassment. As a trial, I asked them to map‑colour a basketweave tiling (which underpins the Cairo tiling) using four colours, making it explicit that no adjacent tiles were to share a colour. I show an example by Grok, which failed miserably; all four AIs share the same broad failing. No “understanding” is shown here whatsoever. Initially, I thus dismissed this outlet.

However, the failure is largely due to my own ignorance about AI. The systems above are all generative AI models, which are wholly unsuitable for this type of problem. I am informed:

Why Grok, Copilot, and Gemini fail. They treat your prompt as a creative pattern‑generation task. They do not build the adjacency graph of the double basketweave tiling. They do not enforce constraints like “no two adjacent tiles share a colour”. They hallucinate colourings because they cannot reason over 64 tiles with adjacency constraints. This is exactly the failure mode you’ve already seen.

No existing off‑the‑shelf generative AI model can reliably solve map‑colouring problems on the double basketweave tiling, because the task requires graph‑theoretic constraint satisfaction, not generative pattern synthesis. But you can get an AI to do it — just not the chat‑style models. You need a constraint‑solving AI, i.e. a SAT/SMT solver or graph‑colouring engine wrapped in a small Python layer.

And here my problems start. I had vaguely heard of a SAT solver and of Python (a programming language), but that is about it as far as knowledge, programming, and interest go. I am further informed:

What does work: a constraint solver. The double basketweave tiling is a finite graph with 64 nodes (in my 8×8 study) and a fixed adjacency structure. This is a textbook case for: – Z3 (SMT solver) – OR‑Tools CP‑SAT (Google) – MiniZinc (high‑level constraint modelling language) – PySAT (SAT solver interface). These tools do understand adjacency constraints and will produce valid colourings instantly. None of these are “chat AIs”. They are AI reasoning engines designed for constraint satisfaction.

I investigated each in turn. All appear to be bona fide solvers. Further, there are differences between SAT and SMT. I was then invited to download Python, which I did. Copilot gave the impression that it was just a paste‑and‑copy code job, and the program would spit out the colourings in seconds. However, I could not get the program running. I remain sceptical of the outcome. Any volunteers? A more in-depth discussion and the code are here:
https://docs.google.com/document/d/1skyexinttJec08uTJcjA3bixBhB9LTZrlgX_sCXi-U0/edit?usp=sharing

PART ONE - ABSTRACT COLOURINGS

I now examine map colourings for 3–8 colours, in varying degrees, depth, and capacities.

Procedure

Quite how best to go about the analysis is a moot point. I originally started with “by colour”, but despite appearing distinct, this method has problems. For example, the two 4-colourings here are apparently distinct, in that when overlaid the colourings do not match, however, when numbers are applied, they are of the same underlying structure.

Therefore, the best policy is by numbers first, with colour transfer thereof, second. But for the most part of the study I have used colour first, which I will retain for now rather than starting again, but duly “allowed” for, as the study has expanded beyond all reason. 

Basics
For the study, I use an 8 × 8 format, with the squares suitably divided in the underlying basketweave tiling. This is sufficient for the number of tiles to reveal the underlying structure without requiring a disproportionate amount of time to colour by hand (or, more accurately, with the fill bucket), as would be the case with, say, an incremental 16 ×16 (or larger) tiling.

Tile Arrangement
The “arrangement” of the tiles is unusual, meeting at both three‑ and four‑valent vertices. The analysis is largely visual, with colours rather than numbers. Although one can represent colours numerically, this lacks the immediacy of colour itself, where the overall “arrangement” is instantly apparent. However, numbers are arguably better for analytic purposes. 

Procedure
An obvious thought is to use a combinatorial approach and find all possibilities admissible and inadmissible, and weed out the failures from the successes. However, in practice, calculating by hand, in which all combinations are shown, including inadmissible instances, is daunting. Vast datasets are involved. Or at least in the way I was proceeding; there may be better, more efficient (shorter) ways. A case in point is the 3-colouring. This has a data set of 1,728 possibilities! And yet there is a more intelligent approach that will yield the (single) colouring almost immediately. Be that as it may, I ignore the intelligent approach in favour of an all-encompassing study that can be applied (in theory at least) to subsequent higher colourings of 4, 5, 6, etc. So how best to go about this task? A natural first observation is that the tiling contains two rectangular tiles subdivided into a square, set at right angles to each other to form a 1×2 unit of four tiles. This can be described as a core unit, and for the most part forms the basis for the study. Simply stated (as I see it), this block must contain the core colour elements, which are then repeated (translated) to give the colouration.

 1 x 2 unit consisting of four tiles

3 Colours

Three Colours - The Dumb Approach

With the methodology as given above, I begin by systematically showing all possibilities of map colouring the above 1 x 2 unit, of which there are 12 admissible instances, thus giving a Set of 12 blocks. It will be seen that, of necessity (of 3-colouring), the colour distribution is always in the ratio 2, 1, 1, e.g. Blue 2, Green 1, Red 1. The equivalent Cairo tiles are shown below the abstract study.

The Combinations
I now examine all the possible combinations in turn, abstractly. For this, I take all possible ways of arranging three blocks from the set of twelve. (A preliminary trial showed that two blocks do not yield an admissible colouring.) This gives a total of 1,728 combinations, of which only one — namely 2, 7, 11 (highligted in red) — produces an admissible colouring. I did not check all of these by hand; possible, yes, but judged impractical, likely requiring days of work. Instead, a retro‑analysis was undertaken, based on a known admissible three‑colouring. As can be seen, this leads to a rather lengthy list, but I felt it best to show it in full rather than present only the first block of 144 combinations (1+…) and the successful case. I have found that attempts to be “clever” can prove a false economy, so I show the combinations in full. 

1+1+1, 1+1+2, 1+1+3, 1+1+4, 1+1+5, 1+1+6, 1+1+7, 1+1+8, 1+1+9, 1+1+10, 1+1+11, 1+1+12
1+2+1, 1+2+2, 1+2+3, 1+2+4, 1+2+5, 1+2+6, 1+2+7, 1+2+8, 1+2+9, 1+2+10, 1+2+11, 1+2+12
1+3+1, 1+3+2, 1+3+3, 1+3+4, 1+3+5, 1+3+6, 1+3+7, 1+3+8, 1+3+9, 1+3+10, 1+3+11, 1+3+12
1+4+1, 1+4+2, 1+4+3, 1+4+4, 1+4+5, 1+4+6, 1+4+7, 1+4+8, 1+4+9, 1+4+10, 1+4+11, 1+4+12
1+5+1, 1+5+2, 1+5+3, 1+5+4, 1+5+5, 1+5+6, 1+5+7, 1+5+8, 1+5+9, 1+5+10, 1+5+11, 1+5+12
1+6+1, 1+6+2, 1+6+3, 1+6+4, 1+6+5, 1+6+6, 1+6+7, 1+6+8, 1+6+9, 1+6+10, 1+6+11, 1+6+12
1+7+1, 1+7+2, 1+7+3, 1+7+4, 1+7+5, 1+7+6, 1+7+7, 1+7+8, 1+7+9, 1+7+10, 1+7+11, 1+7+12
1+8+1, 1+8+2, 1+8+3, 1+8+4, 1+8+5, 1+8+6, 1+8+7, 1+8+8, 1+8+9, 1+8+10, 1+8+11, 1+8+12
1+9+1, 1+9+2, 1+9+3, 1+9+4, 1+9+5, 1+9+6, 1+9+7, 1+9+8, 1+9+9, 1+9+10, 1+9+11, 1+9+12
1+10+1, 1+10+2, 1+10+3, 1+10+4, 1+10+5, 1+10+6, 1+10+7, 1+10+8, 1+10+9, 1+10+10, 1+10+11, 1+10+12
1+11+1, 1+11+2, 1+11+3, 1+11+4, 1+11+5, 1+11+6, 1+11+7, 1+11+8, 1+11+9, 1+11+10, 1+11+11, 1+11+12
1+12+1, 1+12+2, 1+12+3, 1+12+4, 1+12+5, 1+12+6, 1+12+7, 1+12+8, 1+12+9, 1+12+10, 1+12+11, 1+12+12

2+1+1, 2+1+2, 2+1+3, 2+1+4, 2+1+5, 2+1+6, 2+1+7, 2+1+8, 2+1+9, 2+10, 2+1+11, 2+1+12
2+2+1, 2+2+2, 2+2+3, 2+2+4, 2+2+5, 2+2+6, 2+2+7, 2+2+8, 2+2+9, 2+2+10, 2+2+11, 2+2+12
2+3+1, 2+3+2, 2+3+3, 2+3+4, 2+3+5, 2+3+6, 2+3+7, 2+3+8, 2+3+9, 2+3+10, 2+3+11, 2+3+12
2+4+1, 2+4+2, 2+4+3, 2+4+4, 2+4+5, 2+4+6, 2+4+7, 2+4+8, 2+4+9, 2+4+10, 2+4+11, 2+4+12
2+5+1, 2+5+2, 2+5+3, 2+5+4, 2+5+5, 2+5+6, 2+5+7, 2+5+8, 2+5+9, 2+5+10, 2+5+11, 2+5+12
2+6+1, 2+6+2, 2+6+3, 2+6+4, 2+6+5, 2+6+6, 2+6+7, 2+6+8, 2+6+9, 2+6+10, 2+6+11, 2+6+12
2+7+1, 2+7+2, 2+7+3, 2+7+4, 2+7+5, 2+7+6, 2+7+7, 2+7+8, 2+7+9, 2+7+10, 2+7+11, 2+7+12
2+8+1, 2+8+2, 2+8+3, 2+8+4, 2+8+5, 2+8+6, 2+8+7, 2+8+8, 2+8+9, 2+8+10, 2+8+11, 2+8+12
2+9+1, 2+9+2, 2+9+3, 2+9+4, 2+9+5, 2+9+6, 2+9+7, 2+9+8, 2+9+9, 2+9+10, 2+9+11, 2+9+12
2+10+1, 2+10+2, 2+10+3, 2+10+4, 2+10+5, 2+10+6, 2+10+7, 2+10+8, 2+10+9, 2+10+10, 2+10+11, 2+10+12
2+11+1, 2+11+2, 2+11+3, 2+11+4, 2+11+5, 2+11+6, 2+11+7, 2+11+8, 2+11+9, 2+11+10, 2+11+11, 2+11+12
2+12+1, 2+12+2, 2+12+3, 2+12+4, 2+12+5, 2+12+6, 2+12+7, 2+12+8, 2+12+9, 2+12+10, 2+12+11, 2+12+12

3+1+1, 3+1+2, 3+1+3, 3+1+4, 3+1+5, 3+1+6, 3+1+7, 3+1+8, 3+1+9, 3+1+10, 3+1+11, 3+1+12

3+2+1, 3+2+2, 3+2+3, 3+2+4, 3+2+5, 3+2+6, 3+2+7, 3+2+8, 3+2+9, 3+2+10, 3+2+11, 3+2+12

3+3+1, 3+3+2, 3+3+3, 3+3+4, 3+3+5, 3+3+6, 3+3+7, 3+3+8, 3+3+9, 3+3+10, 3+3+11, 3+3+12

3+4+1, 3+4+2, 3+4+3, 3+4+4, 3+4+5, 3+4+6, 3+4+7, 3+4+8, 3+4+9, 3+4+10, 3+4+11, 3+4+12

3+5+1, 3+5+2, 3+5+3, 3+5+4, 3+5+5, 3+5+6, 3+5+7, 3+5+8, 3+5+9, 3+5+10, 3+5+11, 3+5+12

3+6+1, 3+6+2, 3+6+3, 3+6+4, 3+6+5, 3+6+6, 3+6+7, 3+6+8, 3+6+9, 3+6+10, 3+6+11, 3+6+12

3+7+1, 3+7+2, 3+7+3, 3+7+4, 3+7+5, 3+7+6, 3+7+7, 3+7+8, 3+7+9, 3+7+10, 3+7+11, 3+7+12

3+8+1, 3+8+2, 3+8+3, 3+8+4, 3+8+5, 3+8+6, 3+8+7, 3+8+8, 3+8+9, 3+8+10, 3+8+11, 3+8+12

3+9+1, 3+9+2, 3+9+3, 3+9+4, 3+9+5, 3+9+6, 3+9+7, 3+9+8, 3+9+9, 3+9+10, 3+9+11, 3+9+12

3+10+1, 3+10+2, 3+10+3, 3+10+4, 3+10+5, 3+10+6, 3+10+7, 3+10+8, 3+10+9, 3+10+10, 3+10+11, 3+10+12
3+11+1, 3+11+2, 3+11+3, 3+11+4, 3+11+5, 3+11+6, 3+11+7, 3+11+8, 3+11+9, 3+11+10, 3+11+11, 3+11+12
3+12+1, 3+12+2, 3+12+3, 3+12+4, 3+12+5, 3+12+6, 3+12+7, 3+12+8, 3+12+9, 3+12+10, 3+12+11, 3+12+12 

4+1+1, 4+1+2, 4+1+3, 4+1+4, 4+1+5, 4+1+6, 4+1+7, 4+1+8, 4+1+9, 4+1+10, 4+1+11, 4+1+12

4+2+1, 4+2+2, 4+2+3, 4+2+4, 4+2+5, 4+2+6, 4+2+7, 4+2+8, 4+2+9, 4+2+10, 4+2+11, 4+2+12

4+3+1, 4+3+2, 4+3+3, 4+3+4, 4+3+5, 4+3+6, 4+3+7, 4+3+8, 4+3+9, 4+3+10, 4+3+11, 4+3+12

4+4+1, 4+4+2, 4+4+3, 4+4+4, 4+4+5, 4+4+6, 4+4+7, 4+4+8, 4+4+9, 4+4+10, 4+4+11, 4+4+12

4+5+1, 4+5+2, 4+5+3, 4+5+4, 4+5+5, 4+5+6, 4+5+7, 4+5+8, 4+5+9, 4+5+10, 4+5+11, 4+5+12

4+6+1, 4+6+2, 4+6+3, 4+6+4, 4+6+5, 4+6+6, 4+6+7, 4+6+8, 4+6+9, 4+6+10, 4+6+11, 4+6+12

4+7+1, 4+7+2, 4+7+3, 4+7+4, 4+7+5, 4+7+6, 4+7+7, 4+7+8, 4+7+9, 4+7+10, 4+7+11, 4+7+12

4+8+1, 4+8+2, 4+8+3, 4+8+4, 4+8+5, 4+8+6, 4+8+7, 4+8+8, 4+8+9, 4+8+10, 4+8+11, 4+8+12

4+9+1, 4+9+2, 4+9+3, 4+9+4, 4+9+5, 4+9+6, 4+9+7, 4+9+8, 4+9+9, 4+9+10, 4+9+11, 4+9+12
4+10+1, 4+10+2 4+10+3, 4+10+4, 4+10+5, 4+10+6, 4+10+7, 4+10+8, 4+10+9, 4+10+10, 4+10+11, 4+10+12
4+11+1, 4+11+2 4+11+3, 4+11+4, 4+11+5, 4+11+6, 4+11+7, 4+11+8, 4+11+9, 4+11+10, 4+11+11, 4+11+12
4+12+1, 4+12+2 4+12+3, 4+12+4, 4+12+5, 4+12+6, 4+12+7, 4+12+8, 4+12+9, 4+12+10, 4+12+11, 4+12+12 

5+1+1, 5+1+2, 5+1+3, 5+1+4, 5+1+5, 5+1+6, 5+1+7, 5+1+8, 5+1+9, 5+1+10, 5+1+11, 5+1+12

5+2+1, 5+2+2, 5+2+3, 5+2+4, 5+2+5, 5+2+6, 5+2+7, 5+2+8, 5+2+9, 5+2+10, 5+2+11, 5+2+12

5+3+1, 5+3+2, 5+3+3, 5+3+4, 5+3+5, 5+3+6, 5+3+7, 5+3+8, 5+3+9, 5+3+10, 5+3+11, 5+3+12

5+4+1, 5+4+2, 5+4+3, 5+4+4, 5+4+5, 5+4+6, 5+4+7, 5+4+8, 5+4+9, 5+4+10, 5+4+11, 5+4+12

5+5+1, 5+5+2, 5+5+3, 5+5+4, 5+5+5, 5+5+6, 5+5+7, 5+5+8, 5+5+9, 5+5+10, 5+5+11, 5+5+12

5+6+1, 5+6+2, 5+6+3, 5+6+4, 5+6+5, 5+6+6, 5+6+7, 5+6+8, 5+6+9, 5+6+10, 5+6+11, 5+6+12

5+7+1, 5+7+2, 5+7+3, 5+7+4, 5+7+5, 5+7+6, 5+7+7, 5+7+8, 5+7+9, 5+7+10, 5+7+11, 5+7+12

5+8+1, 5+8+2, 5+8+3, 5+8+4, 5+8+5, 5+8+6, 5+8+7, 5+8+8, 5+8+9, 5+8+10, 5+8+11, 5+8+12

5+9+1, 5+9+2, 5+9+3, 5+9+4, 5+9+5, 5+9+6, 5+9+7, 5+9+8, 5+9+9, 5+9+10, 5+9+11, 5+9+12

5+10+1, 5+10+2, 5+10+3, 5+10+4, 5+10+5, 5+10+6, 5+10+7, 5+10+8, 5+10+9, 5+10+10, 5+10+11, 5+10+12
5+11+1, 5+11+2, 5+11+3, 5+11+4, 5+11+5, 5+11+6, 5+11+7, 5+11+8, 5+11+9, 5+11+10, 5+11+11, 5+11+12
5+12+1, 5+12+2, 5+12+3, 5+12+4, 5+12+5, 5+12+6, 5+12+7, 5+12+8, 5+12+9, 5+12+10, 5+12+11, 5+12+12 

6+1+1, 6+1+2, 6+1+3, 6+1+4, 6+1+5, 6+1+6, 6+1+7, 6+1+8, 6+1+9, 6+1+10, 6+1+11, 6+1+12

6+2+1, 6+2+2, 6+2+3, 6+2+4, 6+2+5, 6+2+6, 6+2+7, 6+2+8, 6+2+9, 6+2+10, 6+2+11, 6+2+12

6+3+1, 6+3+2, 6+3+3, 6+3+4, 6+3+5, 6+3+6, 6+3+7, 6+3+8, 6+3+9, 6+3+10, 6+3+11, 6+3+12

6+4+1, 6+4+2, 6+4+3, 6+4+4, 6+4+5, 6+4+6, 6+4+7, 6+4+8, 6+4+9, 6+4+10, 6+4+11, 6+4+12

6+5+1, 6+5+2, 6+5+3, 6+5+4, 6+5+5, 6+5+6, 6+5+7, 6+5+8, 6+5+9, 6+5+10, 6+5+11, 6+5+12

6+6+1, 6+6+2, 6+6+3, 6+6+4, 6+6+5, 6+6+6, 6+6+7, 6+6+8, 6+6+9, 6+6+10, 6+6+11, 6+6+12

6+7+1, 6+7+2, 6+7+3, 6+7+4, 6+7+5, 6+7+6, 6+7+7, 6+7+8, 6+7+9, 6+7+10, 6+7+11, 6+7+12

6+8+1, 6+8+2, 6+8+3, 6+8+4, 6+8+5, 6+8+6, 6+8+7, 6+8+8, 6+8+9, 6+8+10, 6+8+11, 6+8+12

6+9+1, 6+9+2, 6+9+3, 6+9+4, 6+9+5, 6+9+6, 6+9+7, 6+9+8, 6+9+9, 6+9+10, 6+9+11, 6+9+12

6+10+1, 6+10+2, 6+10+3, 6+10+4, 6+10+5, 6+10+6, 6+10+7, 6+10+8, 6+10+9, 6+10+10, 6+10+11, 6+10+12
6+11+1, 6+11+2, 6+11+3, 6+11+4, 6+11+5, 6+11+6, 6+11+7, 6+11+8, 6+11+9, 6+11+10, 6+11+11, 6+11+12
6+12+1, 6+12+2, 6+12+3, 6+12+4, 6+12+5, 6+12+6, 6+12+7, 6+12+8, 6+12+9, 6+12+10, 6+12+11, 6+12+12 

7+1+1, 7+1+2, 7+1+3, 7+1+4, 7+1+5, 7+1+6, 7+1+7, 7+1+8, 7+1+9, 7+1+10, 7+1+11, 7+1+12

7+2+1, 7+2+2, 7+2+3, 7+2+4, 7+2+5, 7+2+6, 7+2+7, 7+2+8, 7+2+9, 7+2+10, 7+2+11, 7+2+12

7+3+1, 7+3+2, 7+3+3, 7+3+4, 7+3+5, 7+3+6, 7+3+7, 7+3+8, 7+3+9, 7+3+10, 7+3+11, 7+3+12

7+4+1, 7+4+2, 7+4+3, 7+4+4, 7+4+5, 7+4+6, 7+4+7, 7+4+8, 7+4+9, 7+4+10, 7+4+11, 7+4+12

7+5+1, 7+5+2, 7+5+3, 7+5+4, 7+5+5, 7+5+6, 7+5+7, 7+5+8, 7+5+9, 7+5+10, 7+5+11, 7+5+12

7+6+1, 7+6+2, 7+6+3, 7+6+4, 7+6+5, 7+6+6, 7+6+7, 7+6+8, 7+6+9, 7+6+10, 7+6+11, 7+6+12

7+7+1, 7+7+2, 7+7+3, 7+7+4, 7+7+5, 7+7+6, 7+7+7, 7+7+8, 7+7+9, 7+7+10, 7+7+11, 7+7+12

7+8+1, 7+8+2, 7+8+3, 7+8+4, 7+8+5, 7+8+6, 7+8+7, 7+8+8, 7+8+9, 7+8+10, 7+8+11, 7+8+12

7+9+1, 7+9+2, 7+9+3, 7+9+4, 7+9+5, 7+9+6, 7+9+7, 7+9+8, 7+9+9, 7+9+10, 7+9+11, 7+9+12

7+10+1, 7+10+2, 7+10+3, 7+10+4, 7+10+5, 7+10+6, 7+10+7, 7+10+8, 7+10+9, 7+10+10, 7+10+11, 7+10+12
7+11+1, 7+11+2, 7+11+3, 7+11+4, 7+11+5, 7+11+6, 7+11+7, 7+11+8, 7+11+9, 7+11+10, 7+11+11, 7+11+12
7+12+1, 7+12+2, 7+12+3, 7+12+4, 7+12+5, 7+12+6, 7+12+7, 7+12+8, 7+12+9, 7+12+10, 7+12+11, 7+12+12 

8+1+1, 8+1+2, 8+1+3, 8+1+4, 8+1+5, 8+1+6, 8+1+7, 8+1+8, 8+1+9, 8+1+10, 8+1+11, 8+1+12

8+2+1, 8+2+2, 8+2+3, 8+2+4, 8+2+5, 8+2+6, 8+2+7, 8+2+8, 8+2+9, 8+2+10, 8+2+11, 8+2+12

8+3+1, 8+3+2, 8+3+3, 8+3+4, 8+3+5, 8+3+6, 8+3+7, 8+3+8, 8+3+9, 8+3+10, 8+3+11, 8+3+12

8+4+1, 8+4+2, 8+4+3, 8+4+4, 8+4+5, 8+4+6, 8+4+7, 8+4+8, 8+4+9, 8+4+10, 8+4+11, 8+4+12

8+5+1, 8+5+2, 8+5+3, 8+5+4, 8+5+5, 8+5+6, 8+5+7, 8+5+8, 8+5+9, 8+5+10, 8+5+11, 8+5+12

8+6+1, 8+6+2, 8+6+3, 8+6+4, 8+6+5, 8+6+6, 8+6+7, 8+6+8, 8+6+9, 8+6+10, 8+6+11, 8+6+12

8+7+1, 8+7+2, 8+7+3, 8+7+4, 8+7+5, 8+7+6, 8+7+7, 8+7+8, 8+7+9, 8+7+10, 8+7+11, 8+7+12

8+8+1, 8+8+2, 8+8+3, 8+8+4, 8+8+5, 8+8+6, 8+8+7, 8+8+8, 8+8+9, 8+8+10, 8+8+11, 8+8+12

8+9+1, 8+9+2, 8+9+3, 8+9+4, 8+9+5, 8+9+6, 8+9+7, 8+9+8, 8+9+9, 8+9+10, 8+9+11, 8+9+12

8+10+1, 8+10+2, 8+10+3, 8+10+4, 8+10+5, 8+10+6, 8+10+7, 8+10+8, 8+10+9, 8+10+10, 8+10+11, 8+10+12
8+11+1, 8+11+2, 8+11+3, 8+11+4, 8+11+5, 8+11+6, 8+11+7, 8+11+8, 8+11+9, 8+11+10, 8+11+11, 8+11+12
8+12+1, 8+12+2, 8+12+3, 8+12+4, 8+12+5, 8+12+6, 8+12+7, 8+12+8, 8+12+9, 8+12+10, 8+12+11, 8+12+12 

9+1+1, 9+1+2, 9+1+3, 9+1+4, 9+1+5, 9+1+6, 9+1+7, 9+1+8, 9+1+9, 9+1+10, 9+1+11, 9+1+12

9+2+1, 9+2+2, 9+2+3, 9+2+4, 9+2+5, 9+2+6, 9+2+7, 9+2+8, 9+2+9, 9+2+10, 9+2+11, 9+2+12

9+3+1, 9+3+2, 9+3+3, 9+3+4, 9+3+5, 9+3+6, 9+3+7, 9+3+8, 9+3+9, 9+3+10, 9+3+11, 9+3+12

9+4+1, 9+4+2, 9+4+3, 9+4+4, 9+4+5, 9+4+6, 9+4+7, 9+4+8, 9+4+9, 9+4+10, 9+4+11, 9+4+12

9+5+1, 9+5+2, 9+5+3, 9+5+4, 9+5+5, 9+5+6, 9+5+7, 9+5+8, 9+5+9, 9+5+10, 9+5+11, 9+5+12

9+6+1, 9+6+2, 9+6+3, 9+6+4, 9+6+5, 9+6+6, 9+6+7, 9+6+8, 9+6+9, 9+6+10, 9+6+11, 9+6+12

9+7+1, 9+7+2, 9+7+3, 9+7+4, 9+7+5, 9+7+6, 9+7+7, 9+7+8, 9+7+9, 9+7+10, 9+7+11, 9+7+12

9+8+1, 9+8+2, 9+8+3, 9+8+4, 9+8+5, 9+8+6, 9+8+7, 9+8+8, 9+8+9, 9+8+10, 9+8+11, 9+8+12

9+9+1, 9+9+2, 9+9+3, 9+9+4, 9+9+5, 9+9+6, 9+9+7, 9+9+8, 9+9+9, 9+9+10, 9+9+11, 9+9+12

9+10+1, 9+10+2, 9+10+3, 9+10+4, 9+10+5, 9+10+6, 9+10+7, 9+10+8, 9+10+9, 9+10+10, 9+10+11, 9+10+12
9+11+1, 9+11+2, 9+11+3, 9+11+4, 9+11+5, 9+11+6, 9+11+7, 9+11+8, 9+11+9, 9+11+10, 9+11+11, 9+11+12
9+12+1, 9+12+2, 9+12+3, 9+12+4, 9+12+5, 9+12+6, 9+12+7, 9+12+8, 9+12+9, 9+12+10, 9+12+11, 9+12+12 

10+1+1, 10+1+2, 10+1+3, 10+1+4, 10+1+5, 10+1+6, 10+1+7, 10+1+8, 10+1+9, 10+1+10, 10+1+11, 10+1+12

10+2+1, 10+2+2, 10+2+3, 10+2+4, 10+2+5, 10+2+6, 10+2+7, 10+2+8, 10+2+9, 10+2+10, 10+2+11, 10+2+12

10+3+1, 10+3+2, 10+3+3, 10+3+4, 10+3+5, 10+3+6, 10+3+7, 10+3+8, 10+3+9, 10+3+10, 10+3+11, 10+3+12

10+4+1, 10+4+2, 10+4+3, 10+4+4, 10+4+5, 10+4+6, 10+4+7, 10+4+8, 10+4+9, 10+4+10, 10+4+11, 10+4+12

10+5+1, 10+5+2, 10+5+3, 10+5+4, 10+5+5, 10+5+6, 10+5+7, 10+5+8, 10+5+9, 10+5+10, 10+5+11, 10+5+12

10+6+1, 10+6+2, 10+6+3, 10+6+4, 10+6+5, 10+6+6, 10+6+7, 10+6+8, 10+6+9, 10+6+10, 10+6+11, 10+6+12

10+7+1, 10+7+2, 10+7+3, 10+7+4, 10+7+5, 10+7+6, 10+7+7, 10+7+8, 10+7+9, 10+7+10, 10+7+11, 10+7+12

10+8+1, 10+8+2, 10+8+3, 10+8+4, 10+8+5, 10+8+6, 10+8+7, 10+8+8, 10+8+9, 10+8+10, 10+8+11, 10+8+12

10+9+1, 10+9+2, 10+9+3, 10+9+4, 10+9+5, 10+9+6, 10+9+7, 10+9+8, 10+9+9, 10+9+10, 10+9+11, 10+9+12

10+10+1, 10+10+2, 10+10+3, 10+10+4, 10+10+5, 10+10+6, 10+10+7, 10+10+8, 10+10+9, 10+10+10, 10+10+11, 10+10+12

10+11+1, 10+11+2, 10+11+3, 10+11+4, 10+11+5, 10+11+6, 10+11+7, 10+11+8, 10+11+9, 10+11+10, 10+11+11, 10+11+12

10+12+1, 10+12+2, 10+12+3, 10+12+4, 10+12+5, 10+12+6, 10+12+7, 10+12+8, 10+12+9, 10+12+10, 10+12+11, 10+12+12 

11+1+1, 11+1+2, 11+1+3, 11+1+4, 11+1+5, 11+1+6, 11+1+7, 11+1+8, 11+1+9, 11+1+10, 11+1+11, 11+1+12

11+2+1, 11+2+2, 11+2+3, 11+2+4, 11+2+5, 11+2+6, 11+2+7, 11+2+8, 11+2+9, 11+2+10, 11+2+11, 11+2+12

11+3+1, 11+3+2, 11+3+3, 11+3+4, 11+3+5, 11+3+6, 11+3+7, 11+3+8, 11+3+9, 11+3+10, 11+3+11, 11+3+12

11+4+1, 11+4+2, 11+4+3, 11+4+4, 11+4+5, 11+4+6, 11+4+7, 11+4+8, 11+4+9, 11+4+10, 11+4+11, 11+4+12

11+5+1, 11+5+2, 11+5+3, 11+5+4, 11+5+5, 11+5+6, 11+5+7, 11+5+8, 11+5+9, 11+5+10, 11+5+11, 11+5+12

11+6+1, 11+6+2, 11+6+3, 11+6+4, 11+6+5, 11+6+6, 11+6+7, 11+6+8, 11+6+9, 11+6+10, 11+6+11, 11+6+12

11+7+1, 11+7+2, 11+7+3, 11+7+4, 11+7+5, 11+7+6, 11+7+7, 11+7+8, 11+7+9, 11+7+10, 11+7+11, 11+7+12

11+8+1, 11+8+2, 11+8+3, 11+8+4, 11+8+5, 11+8+6, 11+8+7, 11+8+8, 11+8+9, 11+8+10, 11+8+11, 11+8+12

11+9+1, 11+9+2, 11+9+3, 11+9+4, 11+9+5, 11+9+6, 11+9+7, 11+9+8, 11+9+9, 11+9+10, 11+9+11, 11+9+12

11+10+1, 11+10+2, 11+10+3, 11+10+4, 11+10+5, 11+10+6, 11+10+7, 11+10+8, 11+10+9, 11+10+10, 11+10+11, 11+10+12
11+11+1, 11+11+2, 11+11+3, 11+11+4, 11+11+5, 11+11+6, 11+11+7, 11+11+8, 11+11+9, 11+11+10, 11+11+11, 11+11+12
11+12+1, 11+12+2, 11+12+3, 11+12+4, 11+12+5, 11+12+6, 11+12+7, 11+12+8, 11+12+9, 11+12+10, 11+12+11, 11+12+12

12+1+1, 12+1+2, 12+1+3, 12+1+4, 12+1+5, 12+1+6, 12+1+7, 12+1+8, 12+1+9, 12+1+10, 12+1+11, 12+1+12

12+2+1, 12+2+2, 12+2+3, 12+2+4, 12+2+5, 12+2+6, 12+2+7, 12+2+8, 12+2+9, 12+2+10, 12+2+11, 12+2+12

12+3+1, 12+3+2, 12+3+3, 12+3+4, 12+3+5, 12+3+6, 12+3+7, 12+3+8, 12+3+9, 12+3+10, 12+3+11, 12+3+12

12+4+1, 12+4+2, 12+4+3, 12+4+4, 12+4+5, 12+4+6, 12+4+7, 12+4+8, 12+4+9, 12+4+10, 12+4+11, 12+4+12

12+5+1, 12+5+2, 12+5+3, 12+5+4, 12+5+5, 12+5+6, 12+5+7, 12+5+8, 12+5+9, 12+5+10, 12+5+11, 12+5+12

12+6+1, 12+6+2, 12+6+3, 12+6+4, 12+6+5, 12+6+6, 12+6+7, 12+6+8, 12+6+9, 12+6+10, 12+6+11, 12+6+12

12+7+1, 12+7+2, 12+7+3, 12+7+4, 12+7+5, 12+7+6, 12+7+7, 12+7+8, 12+7+9, 12+7+10, 12+7+11, 12+7+12

12+8+1, 12+8+2, 12+8+3, 12+8+4, 12+8+5, 12+8+6, 12+8+7, 12+8+8, 12+8+9, 12+8+10, 12+8+11, 12+8+12

12+9+1, 12+9+2, 12+9+3, 12+9+4, 12+9+5, 12+9+6, 12+9+7, 12+9+8, 12+9+9, 12+9+10, 12+9+11, 12+9+12

12+10+1, 12+10+2, 12+10+3, 12+10+4, 12+10+5, 12+10+6, 12+10+7, 12+10+8, 12+10+9, 12+10+10, 12+10+11, 12+10+12
12+11+1, 12+11+2, 12+11+3, 12+11+4, 12+11+5, 12+11+6, 12+11+7, 12+11+8, 12+11+9, 12+11+10, 12+11+11, 12+11+12
12+12+1, 12+12+2, 12+12+3, 12+12+4, 12+12+5, 12+12+6, 12+12+7, 12+12+8, 12+12+9, 12+12+10, 12+12+11, 12+12+12

A thought I had was that perhaps when placed as translated blocks, the (admissible) number sequence would recur; for instance, 2, 7, 11 would be found in blocks 7, 11, 2; 11, 2, 7; 2, 11, 7; 11, 7, 2; 7, 2, 11, but this is not so; all give inadmissible tilings (this was checked by hand). Thus, all combinations here are distinct. Below I give the check of the mooted repeats, beginning with the admissible tiling. 

(a)-(f) (a) 2, 7, 11 (b) 7, 11, 2 (c) 11, 2, 7 (d) 2, 11, 7 (e) 11, 7, 2 (f) 7, 2, 11 

But wait! There are yet more possibilities! All the above are in the form of identical rows. For example
Row 1 1+6+4…
Row 2 1+6+4…
Row 3 1+6+4…

But what if we have mixed, alternate rows? For example
Row 1 1+6+4…
Row 2 2+7+9…
Row 3 1+6+4…
Row 4 2+7+9…

And double up? For example
Row 1 1+6+4…
Row 2 1+6+4…
Row 3 2+7+9…
Row 4 2+7+9….
Row 5 1+6+4…
Row 6 1+6+4…
Row 7 2+7+9…
Row 8 2+7+9…

And so on. The possibilities spiral out of control. 

Although a solitary success has been found, it can hardly be said to be an elegant solution, with a time consuming, brute force approach adopted. What is needed is a smarter approach, which I now address.

Three Colours

Let us begin with colouring an arbitrary patch of contiguous tiles.

Colouration of a contiguous patch of tiles

I begin with red, then green, and then blue is forced. However, the continuation is then not forced. For example, the tiles either side of blue can be coloured with red or green. And so on (not forced) for all the other colours. Although an initial continuation may be found, a distinct possibility is that a conflict may arise. Therefore, I put this patch aside. Instead, I now begin with another 3-colour patch.

Another 3-colour patch

As is immediately apparent, the continuation is forced at each suitable location. For example, in the space above the red tile only a green tile can appear, and in the space below it only a blue tile is allowed. Once these colours are fixed, the remainder of the tiling follows in the same forced manner.

Forced continuation

To more easily see the forced options, instead of colouring the whole tiling, I show an arbitrary patch. For example, the “hole” in the tiling at right can only admit a red tile. Then once that is in place, the other colours are then forced. 

The assembled 3-colouring

From this, it is obvious that no other 3-colouring is possible.

4 Colours

Four colours
Ideally, I would present a smart approach, as above, but I see no simple way to do so. Instead, I adopt a limited “dumb” approach. Having seen the vast dataset for three colours, and the associated combinatorics, the possibilities with four colours expand exponentially. I baulk at the task of fully computing and checking these, but nonetheless I outline a starting point.

As with the dumb approach to 3‑colouring, I use a 1×2 unit, in which I restrict the colour range under the premise of a block of four colours only. This yields a set of 24 blocks. (One might also include the set of 12 three‑colour blocks, with an asymmetric distribution, giving a total set of 36 blocks, but this is judged too complex to compute by hand.)

Combinations
1+1+1+1, 1+1+1+2, 1+1+1+3, 1+1+1+4, 1+1+1+5, 1+1+1+6, 1+1+1+7, 1+1+1+8, 1+1+1+9, 1+1+1+10, 1+1+1+11, 1+1+1+12, 1+1+1+13, 1+1+1+14, 1+1+1+15, 1+1+1+16, 1+1+1+17, 1+1+1+18, 1+1+1+19, 1+1+1+20, 1+1+1+21, 1+1+1+22, 1+1+1+23, 1+1+1+24

1+2+1+1, 1+2+1+2, 1+2+1+3, 1+2+1+4, 1+2+1+5, 1+2+1+6, 1+2+1+7, 1+2+1+8, 1+2+1+9, 1+2+1+10, 1+2+1+11, 1+2+1+12, 1+2+1+13, 1+2+1+14, 1+2+1+15, 1+2+1+16, 1+2+1+17, 1+2+1+18, 1+2+1+19, 1+2+1+20, 1+2+1+21, 1+2+1+22, 1+2+1+23, 1+2+1+24

And so on up to 1+24+24+24. This gives 576 blocks. This then completes the first series

This then continues with the next block

2+1+1+1, and so on, continuing to 24+24+24+24

24+24+24+1, 24+24+24+2, 24+24+24+3, 24+24+24+4,24+24+24+5, 24+24+24+61,24+24+24+7, 24+24+24+8,24+24+24+9, 24+24+24+10,24+24+24+11, 24+24+24+11, 24+24+24+13, 24+24+24+14,24+24+24+15, 24+24+24+16,24+24+24+17, 24+24+24+18,24+24+24+19, 24+24+24+20, 24+24+24+21, 24+24+24+22, 24+24+24+23, 24+24+24+24

24 x 24 = 576. 576 x 24 = 13,824 projected blocks!

And of course, this does not take into account the possibilities of alternate rows and doubling‑up, as discussed for three‑colouring. Analysing the four‑colour case by hand in this way is wholly impractical. A curtailed version is possible, however, and from this I have found eight distinct colourings—an improvement on the three instances previously known from Rinus Roelofs. I give these without comment on the procedure, as it is difficult to set out in a cohesive manner. What I will say is that they were found at the very beginning of the process, which suggests there may be more to come. In fact, many more. But the study has exhausted my patience, and so I leave it there, with the thought that an “instant” enumeration may be achievable with a SAT solver. 

Set of 24

The Set of 9 Distinct 4-Colourations (A-I)
Each instance is accompanied with commentary, to varying degrees.

4-Colouring A (1+1)
No ribbons
Each horizontal hexagon unit cell has a colour distribution in the ratio of 1, 1, 1, 1
Each vertical hexagon unit cell has a colour distribution in the ratio of 1, 1, 1, 1
Yeloow, red, green, and blue appear in one orientation
The simplest and most obvious colouring. This colouring arrangement is shown in Roelofs, Fig. 1c. This is frequently to be seen among the in situ pavings, likely being the most ‘intuitive’ colouring of all, including the 3-colouring, in that one simply places tiles of the same colour in the same orientation, and the colouration repeats as above; no "thought" as such required.

4-Colouring B
Green, blue partial ribbons
Each horizontal hexagon unit cell has a colour distribution in the  ratio of 2, 1, 1
Each vertical hexagon unit cell has a colour distribution in the  ratio of 1, 1, 1, 1
Blue and green appear in two orientations, with yellow and red in one orientation

4-Colouring C
Yellow, red partial ribbons
Each horizontal hexagon unit cell has a colour distribution in the ratio of 1, 1, 1, 1
Each vertical hexagon unit cell has a colour distribution in the ratio of 2, 1, 1
Yellow and red appear in two orientations, with green and blue in one orientation

4-Colouring D
Yellow, red, green, blue partial ribbons
Each horizontal hexagon unit cell has a colour distribution in the ratio of 2, 1, 1
Each vertical hexagon unit cell has a colour distribution in the ratio of 2, 1, 1
Yellow red, green, and blue appear in two orientations

4-Colouring E
No ribbons; yellow, red, green, blue are all islands
Each horizontal hexagon unit cell has a colour distribution in the ratio of 1, 1, 1, 1
Each vertical hexagon unit cell has a colour distribution in the ratio of 1, 1, 1, 1
Yellow and green appear in two orientations, with red and blue in one orientation

4-Colouring F
Yellow and red are partial ribbons
Each horizontal hexagon unit cell has a colour distribution in the ratio of 2, 1, 1
Each vertical hexagon unit cell has a colour distribution in the ratio of 2, 1, 1
Red and yellow appear in two orientations, with blue and green in one orientation 

4-Colouring G (2+3 AR)

Green and blue are infinite ribbons

Each horizontal hexagon unit cell has a colour distribution in the ratio of 1, 1, 1, 1
Each vertical hexagon unit cell has a colour distribution in the ratio of 1, 1, 1, 1

Blue and green appear in two orientations, with yellow and red in one orientation 

4-Colouring H (2+8 AR)

Yellow, red, blue and green are infinite ribbons

Each horizontal hexagon unit cell has a colour distribution in the ratio of 1, 1, 1, 1
Each vertical hexagon unit cell has a colour distribution in the ratio of 1, 1, 1, 1
Yellow, red, blue, and green appear in two orientations

This is shown in Roelofs, Figure 1a

4-Colouring I

No ribbons
Each horizontal hexagon unit cell has a colour distribution in the ratio of 1, 1, 1, 1
Each vertical hexagon unit cell has a colour distribution in the ratio of 1, 1, 1, 1
Yellow, red, blue, and green appear in two orientations

This is shown in Roelofs, Figure 1b

A-F with numbers (to help in analysis for distinctness)

5 Colours
Select five-colourings from the work of George Baloglou.

So far as I am aware, only George Baloglou has looked at five colours! Two examples from our private correspondence are shown above. With the experience gained from the four colours, in which this is barely tractable by hand, I am not disposed to studying this likely even more complex colouration in any way, not even an outline of the possibilities. 

Six Colours
Select six-colouring from the work of George Baloglou.

Once again, so far as I am aware, only George Baloglou has looked at six colours! For a more in-depth analysis of six-colouring possibilities, see Baloglou's blog, with 38 examples (link below). 

As alluded to above, I am not disposed to studying this likely even more complex colouration in any way.

Seven Colours
Select seven-colouring from the work of George Baloglou.

Once again, so far as I am aware, only George Baloglou (2026, Facebook correspondence) has briefly looked at seven colours! 

As alluded to above I am not disposed to studying this likely even more complex colouration in any way.

Eight Colours

Although at first thought eight colours would be horrendously complex, there is at least one simple example, with the colour arrangement of all ribbons. With the likelihood of complexity, no further colourings were studied. Derived from a Math Overflow posting on the Cairo tiling, although there are certain ambiguities about the post, an eight-colouring is described and recreated above.

POLYCAIRO COLOURING (MOSTLY SELECT)

(i) Dicairos (2/2 possibilities)
(ii) Tricairos (1/5 possibilities)

(iii) Tetracairos (1/17 possibilities)

Another possibility is polycairo colouring, where adjacent tiles are combined, resulting in a non-map colouring. The enumeration of these has been determined and illustrated: monocairo (1), dicairos (2), tricairos (5) tetracairos (17), pentacairos (55), and so on (OEIS gives a longer listing). In effect, these blocks can be coloured (non-map-colouring) and so transferred as a tiling, in theory at least. However, much to my surprise, according to a brief search and AI, the tiling possibilities of these have not been studied! Ideally I would study this to my normal thorough nature, but I am more than a little wearied at all this (now two months into the study), and so largely leave this unresolved. Instead, I show a few arbitrary examples from the first few in the sequence, namely the two dicairos (one of which I name the Macmillan configuration), tricairos and tetracairos. A few of these colourings are to be seen in the in-situ pavings - see the next part.

DICAIRO A (MACMILLAN)
Using two, three and four colours

DICAIRO B
Three possibilities,  using three and four colours

TRICAIRO
Using three colours

TETRACAIROS

COMBINING (HYBRID)

I now investigate the possibility of combining both types, i.e., map colouring and Macmillan, with what I term a hybrid. It will soon be seen that, as with map colourings in general, the study expands beyond all reason, so much so that I only pay lip service to it and only show a few arbitrary examples for three and (one) instance of four colours. As much as I would like to determine “all hybrid possibilities”, it is judged too time-consuming for its intrinsic worth.

Three colours, Combination A

Three colours, Combination B

Four colours, Combination A

Stripes
This instance has been seen in the situ pavings, in Heliopolis, outside Caesar’s Place hotel, 1994, although it is derived from a drawing by Robin Wilson. It is not known if the paving is still extant.

Various Published Coloured Diagrams in Books and Articles

An observation is just how scant (and old!) the literature on the topic is — and what does exist is decidedly lightweight. “Primary” sources include Tomlinson (1963), who shows the simplest four-colouring; Coxeter (1963) includes a single three‑colouring on a book cover,;Loeb (1971) also shows the simplest four-colouring, while Roelofs (1998) shows three four‑colour diagrams. “Secondary” sources offer only Shubnikov and Belov’s (1964) four‑colour basketweave tilings. Beyond these few fragments, there is nothing! Unbelievably, it's over a quarter of a century since the last (lightweight) study! 

Primary

J. W. Tomlinson, Do it Yourself, August 1963

Four-colouring

H. S. M. Coxeter. Regular Complex Polytopes, 1963
Three-Colouring 

Arthur L. Loeb, Colour and Symmetry, 1971

Four-colouring

Rinus Roelofs, Pythagoras, 1998
Three, Four-colourings
For unclear reasons, Roelofs uses two very dark colours, resulting in a “hard-to-see” arrangement.

Secondary

A. V. Shubnikov and N. V. Belov, Coloured Symmetry, 1964

Two basketweave colourings

PART 2

In Situ Colourings

Having examined the colouring possibilities in the abstract, I now turn to the in situ pavings to see how these ideas were incorporated—if at all. Here I exclude the square format, as these are not of single tiles. A persistent question is whether homeowners were given any guidance, perhaps in the form of a leaflet showing possible arrangements, or whether they were simply left to their own devices. The evidence is inconclusive. One might expect some form of instruction, yet the complete absence of the minimal three‑colour map‑colouring solution—remarkable in itself—suggests that no such guidance was supplied. Without it, a three‑colouring is far from straightforward for the typical non‑mathematical homeowner.

Even so, it stands to reason that most people would look for a pattern rather than tolerate a jarring or disorderly arrangement. By far the most common scheme is a four‑colouring following the rule “same orientation, same colour”, likely because of its sheer ease of placement: one simply keeps track of orientation and assigns colour accordingly. Whether a layperson would intuit this rule unaided is another matter. If a picture were provided at the point of sale, that could explain its relative frequency. If not, the irregular patches of contiguous colour seen in some pavings may reflect either lack of interest or frustration after unsuccessful attempts to impose regularity, leading to tiles being placed “any which way”.

The sole known document—the advertisement—is frustratingly unhelpful. It shows only a small monochrome patch of the tiles that is difficult to interpret. Even the number of colours is unclear. What can be discerned, however, is that it does not follow the “same orientation, same colour” rule, as the white tiles fail to conform to it.

Map Colourings

3 Colours. No known examples.

4 Colours. Two “types” are seen, although widely differing in popularity. “Same orientation, same colour” is overwhelmingly favoured, where “Infinite Ribbons” has only two known instance. The terms are explained below.

Maadi © Gregg De Young

“Same orientation, same colour”. This is by far the most frequently seen colouration, likely due to its sheer ease of placement, in that to achieve this all one needs to consider and follow when placing the tiles is a “same orientation, same colour” rule. However, again, one would have to ask if the layman would be aware of this. If there were a picture provided on supply, then this could explain its relative frequency. Possibly not, which would explain instances of contiguous colour, likely out of either lack of interest or lack of success with trying to repeat in a regular way, leading to frustration, and so placing the tiles “any which way”.

This is shown abstractly as example A in the above 4-colour listings.

Maadi, © Robert Fathauer
Infinite ribbons. This arrangement is far less common, with only two documented examples — one in Maadi and another at the Gezira Sporting Club. Its arrangement is straightforward enough for a homeowner to lay, but only with foresight, as a certain amount of planning is required.
It appears abstractly as example G in the listings above.

“Infinite Ribbons”, Gezira Sporting Club, © Michel Hébert

Among the polycairo arrangements, the “Macmillan” configuration is seen almost to the exclusion of all other possibilities. It appears widely, typically in a variety of two‑colourings. Two examples are shown, black and white and burgendy and ochre. Again, this is intuitively one of the simpler arrangements to envisage, hence likely its widespread usage. Higher‑order instances — those of three or four colours — are not observed, with one possible exception.

© Gregg De Young, American University in Cairo

Terracing at the Old Cataract Hotel, Aswan. The sighting is no longer extant. 

Location unknown © Pam Garnett

This sighting is somewhat difficult to interpret, for a variety of reasons. First, only a relatively small patch of the tiles is shown. Second, seemingly a Macmillan configuration of three colours was originally intended, where possibly the homeowner got confused and placed the foreground tiles incorrectly (as a tetracairo). Whatever, a structured colouring is obviously intended.

Hybrid

Some kind of structure is attempted here, but due to the raking angle of the picture, sun and shadows, it is not entirely clear what the homeowner is striving for. For unclear reasons, the tiles in the foreground do not match with the more ordered arrangement in the distance.

Maadi driveway, © Gregg De Young

Others

A strip arrangement in Heliopolis outside Caesar's Palace Hotel, in 1994, was reported by Robin Wilson, in a lecture at Gresham College, 2008, but was not photographed. This is shown abstractly in the listings above.

Acknowledgements

George Baloglou, for many helpful discussions on colouring matters over many years.

John Sharp, for supplying the Coxeter front cover.

Bibliography

Baloglou, George. “The Cairo tiling in six colors”. Crystallomath (blog). October 18, 2013.

An extensive discussion on 6-colouring.
http://crystallomath.wordpress.com/2013/10/18/cairo-six/

Coxeter, H. S. M. Regular Complex Polytopes. Cambridge University Press, Second edition, 1963.

The front cover shows a single 3-colouring. Only the second edition has the tiling as the cover.

The dating of the second edition varies between sources, with some listing 1961 and others 1963. This discrepancy arises from mid‑century printing practices: early impressions were produced in 1961, but the formally issued edition carries a copyright date of 1963. As Copilot notes: Coxeter’s second edition was printed in 1961 but formally issued with a copyright date of 1963; the standard citation uses 1963.

Grünbaum, Branko and G. C. Shephard. Tilings and Patterns. W. H. Freeman, 1987.

P. 467 shows a coloured double basketweave tiling derived from Belov, of the same orientation/placement 4-colour type.

Loeb, Arthur L. Color and Symmetry. New York, Wiley-Interscience, 1971

Colour plates at the end of the book (unpaginated) show a single Cairo tiling of the simplest four-colouring.

Macmillan, Robert. 'Pyramids and pavements: some thoughts from Cairo', Mathematical Gazette, December 1979, pp. 251–255.

Gives the one fused colouring, which I title as “Macmillan” in his honour.

Roelofs, Rinus. “Tegels kleuren” (Colouring Tiles). Pythagoras, Vol. 37, No. 4, April 1998, pp. 22–23.

Gives three four-colourings.

https://pyth.eu/uploads/user/ArchiefPDF/Pyth37-4.pdf

Shubnikov, A. V. and N. V. Belov. Colored Symmetry. Pergamon Press, 1964.

Plate I gives two colourings of the double basketweave tiling.

https://archive.org/details/a.-v.-shubnikov-n.-v.-belov-colored-symmetry-1964/page/n24/mode/thumb

Tomlinson, J. W. Do it Yourself, August 1963

Pp. 900–901 gives a colour-coded key diagram of the simplest four-colouring.

https://abarothsworld.com/Puzzles/Polyforms/Polycairos.htm

Gives a census of the polycairos.

Page History
3 July–16 October 2026. Wholesale changes were made to the original page (archived at the Wayback Machine), although the same broad format was retained, of abstract colouring followed by the in situ sightings. Notably, the presentation of the abstract colourings changed to 8 x 8 diagrams. As well as showing  the underlying basketweave tiling side by side. In general, I have been more expansive. For example, previously I showed only three, four-colourings, whereas the return now shows eight, along with analysis. Also, I expanded the range, from 7- to 8 four-colourings. I now have a distinct polycairo section, where it was previously titled “fused colourings”, along with additional examples. Similarly, I have improved on the in situ colourings, with more expansive text. Finally, I have included “end matters” in the form of a bibliography and acknowledgements. In short, each aspect has been expanded for the better. 

17–18 July 2025. New Sites 'adjustments'. The conversion had left the page with various shortcomings in presentation. This included large spaces between texts, captions not attached to pictures, inconsistent text boxes, with titles and discussion disjoint. All are now corrected. All the usual errors were corrected in Grammarly, pending a more extensive review later.  An attempt was made at a more consistent text style, resulting in a slight improvement, albeit marrying different aspects was not straightforward.  Nonetheless, the page is broadly ordered.

Created: 8 February 2013. Last update 21 October 2013

A single instance has been found, but the properties of the colouring remain uncertain, due to a small patch, from which one cannot tell if it is regular or not. Upon examination, isolating each colour to see if there is regularity, it would appear not. However, for reasons as above, this is a provisional statement, subject to change. As few tiles are shown, certainly not enough to determine if the pattern continues, whether this is a regular colouring is uncertain. Upon preliminary investigation, it would appear not. If a regular five colouring was indeed discerned, this would be significant, as it would suggest a mathematician is behind this; beyond all reasonable doubt, a lay person would not go to the time and trouble for such an involved colouring.