One great problem, three kinds of immersion
From a story to your own conjecture
Problem stories open curiosity. The problem workshop is the next place, where curiosity becomes a testable idea.
- 1 · DiscoverYou are hereMeet the questionLearn when it appeared and why it still holds people’s attention.
- 2 · ChallengeTest one caseMake your first observation in five minutes with a drawing, calculation, or colors.
- 3 · DevelopBuild on ideasPublish an observation and grow it through comparisons, conjectures, and counterexamples.
Hadwiger–Nelson — How Many Colors Can Guard the Entire Plane?
Open for 76 years since 1950. Points exactly one unit apart must have different colors. The answer is still one of only 5, 6, or 7.
Test small cases immediately with a drawing, divisors, or colored pencils.
The starting level measures how easily you can understand and test small cases. It is not the difficulty of a complete proof.
Jump to your first five minutesSee the puzzle visually
The lower bound jumped from 4 to 5 in 2018. The goal now is to eliminate or confirm one of the three survivors: 5, 6, or 7.
Problem statement
Color every point of the plane so that any two points exactly one unit apart receive different colors. What is the minimum number ?
The story of this puzzle
Edward Nelson first considered the question as a Princeton student in 1950. The plane has infinitely many points, but the rule is only this: unit-distance neighbors cannot share a color. The small Moser spindle shows that three colors fail, while a hexagonal construction shows seven suffice. For decades the answer was trapped among 4, 5, 6, and 7.
In 2018, an unexpected breakthrough arrived. Aubrey de Grey, better known for research on biological aging, built a 1,581-vertex unit-distance graph that cannot be colored with four colors. That raised the lower bound to five; collaborators later found smaller witness graphs. The exact answer remains 5, 6, or 7.
This problem is not locked behind specialist notation. The moment you draw points, join unit-distance pairs, and get stuck coloring the resulting graph, human pattern sense meets computer search. De Grey’s breakthrough likewise grew from assembling small obstructions into a larger trap for four colors.
Try it yourself
Colored-pencil challenge. Join two equilateral triangles into a rhombus and find the fewest colors under the unit-distance rule. Then try to 3-color the Moser spindle. Explain the exact point where a fourth color becomes unavoidable: that explanation is a small lower-bound proof.
Beyond MathVoyage
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