Minimum odd-covered area of congruent disk configurations
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For an odd number of unit disks in the plane, how small can the area covered an odd number of times be? Is there any positive universal lower bound, and does the proposed value π hold?
Why it matters and what counts as progress
Why it matters: The question tests whether parity coverage forces a nonzero area regardless of overlap.
What counts as progress: A configuration with rigorously bounded smaller odd-covered area, or a universal lower bound. Monte Carlo area estimates identify candidates but do not certify an exact counterexample.
How the problem developed
- 6 Jun 2026 · Counterexample [1]
Stefan Steinerberger Gave a configuration of seven unit disks whose odd-covered area is at most 3.12, disproving the proposed lower bound π.
AI contribution
A Remark on the Odd Area of Unit Disks
AlphaEvolve found 51- and 151-unit-disk configurations whose odd-covered areas are estimated by Monte Carlo calculations at about 2.73 and 2.51, respectively.
Sources
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