## Fall 2016, problem 21

Determine all values of $n$ such that it is possible to divide a triangle into $n$ smaller triangles such that there are not three collinear vertices and such that each vertex belongs to the same number of segments.

### Comments

What about the figure in the theorem of Morley ?

edit: The possibilities are:[You get these figures, by taking the surface of a tetrahedron, octahedron or icosahedron, deleting one face and then flattening out the rest. As an aside, the other two platonic solids would give you a subdivision of a quadliteral into smaller quadliterals (cube) or of a pentagon into smaller pentagons (dodecahedron). Both subdivisions satisfy the rest of the problem requirements.]

Here's what I wrote before. It's wrong:Notation:After subdivision...Here's a formal

Proof:Euler's formula for planar graphs says: $$ n + v - e = 1 \quad (Eq.1)$$

We sum over all segments the number of vertices on that segment (the latter is always 2). The result is $2e$, of course. Counting this way, we count every vertex $k$ times from the definition of $k$. Therefore:

$$2e = k v \quad (Eq.2)$$

edit: We sum over all triangles the number of vertices of that triangle (the latter is always 3). The result is $3 n$, of course. Counting this way, we count every vertex of the original triangle $(k - 1)$ times and we count all other vertices $k$ times (the difference between these two types of vertices comes from the fact, that the outside of the original triangle is not a triangle). Therefore:$$3 n = (k - 1) \times 3 + k (v - 3) \quad (Eq.3)$$

Here's what I wrote before. It's wrong:Putting together Equations 1 through 3 yields:

$$n (6 - k) = 5 k - 6$$.

The solutions of this equation by positive integers are $(k,n) \in \{(3,3), (4,7), (5,19), (2, 1)\}$. The solution (2,1) does not correspond to a valid subdivision, since in the problem it is specified, that after subdivision the triangles be

smaller.Thanks to rubs, who found a serious bug in my original answer and who brought the morley figure to my attention.

Here's an illustration of the $n=19$ case: