Paving Slabs: The four-colour problem

Suppose you wanted a patio laid but you didn’t want any paving slabs that had adjacent sides to be of the same colour. What is the minimum number of colours that you could achieve this effect with for any possible arrangement of slabs? The answer is surprisingly low: just four. This problem was known as the four-colour problem and troubled mathematicians for over a century until it was eventually proved with the help of computers in the 1970s. 

Two colours are enough for the simplest arrangements: a garden path of alternating slabs or a chessboard where squares of the same colour only touch at the corners. Three colours would suffice in nearly all other situations but some more complex configurations require four different colours. It has been mathematically proven that even the most elaborate crazy paving would never need more than four different colours to achieve the effect of having no adjacent slabs of the same colour.

This conjecture was first formulated (with regards to maps, not patios, but the principle is the same for any two-dimensional plane broken into blocks of colour) in 1850 but not proven until 1976. So why did the problem take so long to solve? The number of possible arrangements of slabs of any shape is basically infinite. It only takes one of those configurations to require five colours to disprove the conjecture. Using many thousands of hours of computer time, the mathematicians Kenneth Appel and Wolfgang Haken were able to reduce the infinity of individual arrangements to just 1,936 “reducible configurations”, or designs from which all other configurations could be deduced, that could then be checked individually to ensure that none needed more than four colours.

Of course, you can have as many or as few colours as you like on your patio. We are pavers, not mathematicians. All our staff at Knight’s are experienced and skilled patio pavers. Our installation method involves digging a deep foundation that is filled with wet cement. Your stonework will be laid on top of this firm base, ensuring that your patio will remain level and true for many years to come. Consult with us about paving slabs or have a glance through our gallery for inspiration. 

For any information on patios, driveways and landscaping projects, don’t hesitate to contact our dedicated team of experts who are ready to help at every stage. You can call us on 01603 360548 or drop us an email at enquiries@knightslandscaping.co.uk