When Paulo turned 15, he invited 43 friends to a party. The cake was in the shape of a regular 15-sided polygon and had 15 candles on it. The candles were placed in such a way that no three candles were in a straight line. Paulo divided the cake into triangular pieces where each cut connected two candles or connected a candle to a vertex. Moreover, no cut crossed another already made. Explain why, by doing this, Paulo was able to give a piece of cake to each of his guests, but he himself was left without any.

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Solution

Let $n$ be the number of triangles into which the cake can be divided under the given conditions. We will sum the interior angles of these triangles in two ways:

- On the one hand, since each triangle has an internal angle sum of $180^{\circ}$, the sum of all the interior angles is $180^{\circ} \cdot n$.
- On the other hand, each interior angle of a triangle is associated with a candle or vertex of the cake. The sum of the interior angles associated with the candles is $360^{\circ} \cdot 15$ and the sum of the interior angles associated with the vertices coincides with the sum of the interior angles of the polygon, which is $180^{\circ} \cdot(15-2)$. Thus, the total sum is also equal to

$$
360^{\circ} \cdot 15+180^{\circ} \cdot(15-2)
$$

Therefore,

$$
180^{\circ} \cdot n=360^{\circ} \cdot 15+180^{\circ} \cdot(15-2)
$$

which implies $n=43$. Since there are 43 guests, Paulo is left without a piece!