42
Curves with three holes · 42 shapes
The 42 combinatorial types of stable curves of genus 3: every way a three-holed curve can pinch into pieces joined at points, counted up to shape. Also the 42 cells of the moduli space of tropical curves of genus 3 (OEIS A174224, a(3) = 42).
Picture a smooth shape with three holes, like a pretzel. Mathematicians call its curve genus 3. Now let it pinch: a hole can shrink to a point, or the shape can split into pieces that touch at single points. How many different patterns can the pieces make? They have all been counted, and the answer is exactly 42. Melody Chan checked it with a computer and wrote it plainly: the tropical moduli space of genus 3 has 42 cells. With two holes there are only 7 patterns. With four holes there are 379. Three holes gives 42.
What it supports
The 42 is printed by the sources, not added up by us. Chan, Combinatorics of the tropical Torelli map (Algebra and Number Theory 6, 2012; arXiv 1012.4539), Theorem 2.12 (i), page 8, says the moduli space of tropical curves of genus 3 has 42 cells. She lists the sequence 0, 0, 7, 42, 379, 4555 as an independent check of work first done by Maggiolo and Pagani. Schwegele (arXiv 2511.15858), Proposition 2.38, states: "There are exactly 42 combinatorial types of stable curves of genus 3." OEIS A174224 gives a(3) = 42, the number of homeomorphism classes of stable curves of genus g. A non-hyperelliptic curve of genus 3 is a plane quartic, a curve of degree 4 drawn in the plane. Deligne and Mumford filled in the space of all such curves by adding the pinched ones, and these 42 types are the pieces of that filled-in space. The fifth Catalan number and p(10) are also 42 as one term of a counting sequence. Same kind of count, different objects.
What it does not
Not a number anyone chose. Not a sign that genus 3 is special because of 42. Not true for other genera (7 for two holes, 379 for four). Not the Hurwitz 1/42 already in the house; that is a different object. Not physics, and tropical here is a kind of geometry, not weather or the zodiac. Not Adams.
ExactnessExact
Form42
Unittypes
Strengthhard
When2010-12-21
Hung24 October 2026