4. When written in ascending order, the nine internal angles from three particular triangles form a sequence where the difference between any adjacent pair of numbers in the sequence is a constant $d$. One of the angles measures $42^{\circ}$. Find all possible values of the size of the largest of the nine angles.
SolUTION
Let $m$ be the median of the nine angles. We can express the nine angles (in degrees) as
$$
m-4 d, m-3 d, m-2 d, m-d, m, m+d, m+2 d, m+3 d, m+4 d .
$$

As the sum of internal angles in each triangle is $180^{\circ}$, these nine numbers add up to $3 \times 180=540$. Hence, $9 m=540$, so $m=60$.
We know that one of the numbers $m-4 d, m-3 d, m-2 d, m-d$ is equal to 42 . Consider each of these four cases in turn:
- Case $1(m-4 d=42)$. This yields $d=4.5$, and angle measures of
$$
42^{\circ}, 46.5^{\circ}, 51^{\circ}, 55.5^{\circ}, 60^{\circ}, 64.5^{\circ}, 69^{\circ}, 73.5^{\circ}, 78^{\circ} \text {. }
$$

We can easily construct three triangles, with their angles, respectively, $\left(42^{\circ}, 64.5^{\circ}, 73.5^{\circ}\right)$, $\left(46.5^{\circ}, 55.5^{\circ}, 78^{\circ}\right)$ and $\left(55.5^{\circ}, 60^{\circ}, 64.5^{\circ}\right)$.
- Case $2(m-3 d=42)$. This yields $d=6$, and angle measures of
$$
36^{\circ}, 42^{\circ}, 48^{\circ}, 54^{\circ}, 60^{\circ}, 66^{\circ}, 72^{\circ}, 78^{\circ}, 84^{\circ} \text {. }
$$

We can easily construct three triangles, with their angles, respectively, $\left(36^{\circ}, 66^{\circ}, 78^{\circ}\right)$, $\left(42^{\circ}, 54^{\circ}, 84^{\circ}\right)$ and $\left(48^{\circ}, 60^{\circ}, 72^{\circ}\right)$.
- Case $3(m-2 d=42)$. This yields $d=9$, and angle measures of
$$
24^{\circ}, 33^{\circ}, 42^{\circ}, 51^{\circ}, 60^{\circ}, 69^{\circ}, 78^{\circ}, 87^{\circ}, 96^{\circ} \text {. }
$$

We can easily construct three triangles, with their angles, respectively, $\left(24^{\circ}, 69^{\circ}, 87^{\circ}\right)$, $\left(33^{\circ}, 51^{\circ}, 96^{\circ}\right)$ and $\left(42^{\circ}, 60^{\circ}, 78^{\circ}\right)$.
- Case 4 ( $m-d=42$ ). This yields $d=18$, which is impossible, as $m-4 d=-12<0$. In conclusion, the only possible values for the size of the largest angle are $78^{\circ}, 84^{\circ}$ and $96^{\circ}$.