Chapman's Problem

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Chapman's Problem is an as-of-yet-unsolved math problem relating to set theory and patterns. Originally proposed in 2004 by then Duke University student Jeremy Chapman, Chapman's Problem seeks a formulaic, mathematical answer to the question:

"How many unique arrangements are possible using n non-overlapping circles?"

For example, when one circle is present, only 1 unique state exists (see illustration below). When two circles are present, 2 unique arrangements exist: the circles may be arranged side by side or one inside the other. When three circles are present, 4 arrangements are possible. However, beyond three circles, the apparent pattern of doubling breaks down and greater than 2x the previous number of arrangements are possible.

Chapman Numbers are defined as the successive numbers of ways to arrange n circles. The first five Chapman Numbers (for n = 1-5) are thus: 1, 2, 4, 9 and 20.

It has been offered that Chapman's Problem is a special case of Catalan's Problem, posed and solved by the Belgian Mathematician Eugène Charles Catalan.

Despite its deceptively simple presentation, no rigorous solution for Chapman's Problem has ever been presented and approved.

See Also

Catalan's Problem

References

Catalan's problemJeremy Chapman's Blog Entry About Chapman's Problem