### Solution to Problem #2

Bojan Basic of Novi Sad (Serbia) showed that there are 18 ways to cut a triangle with three cuts and found a lower bound (which he believes is sharp) of 136 ways to cut a triangle with four cuts. His solution is below.

```We will prove (by counting) that there are 18 essentially different ways of
cutting a triangle with three cuts, and give a lower bound of 136 ways if
four cuts are allowed.

Three cuts:

We shall follow the pictures from the statement, adding one cut to the each
picture in all the possible ways (taking care of what's repeating).

If we have two cuts which do not intersect, and if they connect the same
pair of sides (as in the first picture): the third cut may also connect
the same pair of sides, and in that case it can be added in 9 ways, 4 of
which are essentially different; if the third cut connects one of these
sides with the third one, it can be added in 3 ways, all of which are
essentially different.
```

```If we don't have the previous case, but we do have two cuts which do not
intersect (as in the second picture): the third cut may connect the same
pair of sides as one of the first two cuts already connects, and in that
case it must intersects with it (in order to not repeat what was counted
in the previous paragraph), which leaves only 2 essentially different ways;
if the third cut connects the remaining pair of sides, it can be added in 3
different ways.
```

```This leaves the case when all the cuts intersect. If all of them are
connecting the same pair of sides (we are adding one cut to the third
picture), that gives 2 ways; if all of them are connecting the different
pairs of sides (we are adding one cut to the fourth picture), that gives 2
ways; if exactly two of them are connecting the same pair of sides (for
example, the third picture again), that leaves 2 final ways.
```

```This makes a total of 18 ways.

Four cuts:

The lower bound for four cuts was found by computer, by randomly cutting
the triangle, representing the picture as graph (with meeting points being
the vertices, and the segments joining them being the edges), and checking
how many non-isomorphic are among them. After about 40.000 tries, a total
of 135 cuts was found, and it was standing still till about 55.000 tries
(when the program was stopped). Checking the result showed that the cases
with all four cuts connecting the same pair of sides were taking
particularly long to be found, and then modifying the code in order to
specifically search for only this kind of configuration found one more
(which was found almost immediately, and lasted as the only one for about
another 20.000 tries, when the program was stopped), making a total of
136. Of course, there is no guarantee that these are all of them (although
this heuristic argument is pretty solid), but this indeed is a lower
bound. The 136 graphs are here (as a 754 KB zipped file). For programming,
Mathematica was used.
```

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