Kobon Triangle Problem

The Kobon triangle problem is an unsolved problem in combinatorial geometry first stated by Kobon Fujimura (1903-1983). The problem asks for the largest number ''N''(''k'') of nonoverlapping triangles whose sides lie on an arrangement of ''k'' lines. Variations of the problem consider the projective plane rather than the Euclidean plane, and require that the triangles not be crossed by any other lines of the arrangement..

Known upper bounds

Saburo Tamura proved that the number of nonoverlapping triangles realizable by $k$ lines is at most $\lfloor k(k-2)/3\rfloor$. G. Clément and J. Bader proved more strongly that this bound cannot be achieved when $k$ is congruent to 0 or 2 (mod 6). The maximum number of triangles is therefore at most one less in these cases. The same bounds can be equivalently stated, without use of the floor function, as:
$\begin \frac 13 k (k-2) & \text k \equiv 3,5 \pmod; \\ \frac 13 (k+1)(k-3) & \text k \equiv 0,2 \pmod; \\ \frac 13 (k^2-2k-2) & \text k \equiv 1,4 \pmod. \end$

Solutions yielding this number of triangles are known when $k$ is 3, 4, 5, 6, 7, 8, 9, 13, 15 or 17.Ed Pegg Jr. on Math Games
/ref> For ''k'' = 10, 11 and 12, the best solutions known reach a number of triangles one less than the upper bound.

Known constructions

Given a perfect solution with ''k''₀ > 3 lines, other Kobon triangle solution numbers can be found for all ''k_i''-values where
$k_ = 2\cdot k_ - 1,$
by using the procedure by D. Forge and J. L. Ramirez Alfonsin. For example, the solution for ''k''₀ = 5 leads to the maximal number of nonoverlapping triangles for ''k'' = 5, 9, 17, 33, 65, ....

Examples

Image:KobonTriangle_3.svg, 3 straight lines result in one triangle
Image:KobonTriangle_4.svg, 4 straight lines
Image:KobonTriangle_5.svg, 5 straight lines
Image:KobonTriangle_6.svg, 6 straight lines
Image:KobonTriangle_7.svg, 7 straight lines

See also

*Roberts's triangle theorem, on the minimum number of triangles that $n$ lines can form

References

{{Reflist

External links

*Johannes Bader
"Kobon Triangles"

Discrete geometry
Unsolved problems in geometry
Recreational mathematics
Triangles