Mathematics · the experiment
Colour without sums
One rule: within a colour, never x, y and x + y. Hit the small Schur numbers yourself, then fold the 537 record onto itself.
1 · Colour with two colours
Beginner
Red already holds 3. Which of these numbers can no longer be red?
Two colours. Colour 1, 2, 3 and 4 so that no colour holds x, y and x + y. Tap a number to change its colour.
2 · Move up to three colours
Intermediate
With the recipe S(k + 2) ≥ 10 S(k) + 2, what bound does the new 537 give for eight colours?
Three colours, from 1 to 13. Colour them all without a forbidden triple.
3 · Fold the record
In the record colouring, every number x has the same colour as 538 − x. The number 90 has colour 2.
Expert
Which other number must have colour 2?
4 · Move on to triangles
Colour the link between two points i and j by the colour of |i − j| in a sum-free colouring: no triangle has all three sides the same colour. The article deduces R₆(3) ≥ 539.
Expert
How many points does the network coloured this way with the record have?
Colours held
—