The quantitative Monsky problem asks how nearly an odd number of triangles in a dissection of a unit
square can have equal areas. For a dissection 
into 
nondegenerate triangles with areas 
,
define
 |
(1)
|
and let
 |
(2)
|
The infimum is attained for every 
. Monsky's theorem (Monsky 1970) implies that 
for every fixed odd 
.
The dissections illustrated above contain five, seven, and nine triangles. The labels 
and 
indicate the larger and smaller areas,
respectively. Li (2026) proved the exact value
 |
(3)
|
Every minimizing dissection into five triangles has three triangles of area 
and two triangles of area 
.
For seven triangles,
 |
(4)
|
where 
is the unique root in 
of
 |
(5)
|
Every minimizing dissection into seven triangles has four triangles of area 
and three triangles of area 
.
Li (2026) also constructed a dissection into nine triangles proving
 |
(6)
|
where 
is the unique root in 
of
 |
(7)
|
This construction has five triangles of area 
and four triangles of area 
.
The value 
is the exact minimum within the tilted-strip construction, but the exact value of
remains open.
Li (2026) reports that OpenAI GPT models accessed through ChatGPT and Codex contributed substantially to refining the research program, exploring constructions, developing proof decompositions, and designing and auditing the verification code. The author selected and checked the final statements, arguments, and code and assumes responsibility for the results.
