60.5. Sharon has a chandelier containing $n$ identical candles. She lights up the candles for $n$ consecutive Sundays in the following manner: the first Sunday, she lights up one candle for one hour; the second Sunday, she lights up two candles, conveniently chosen, for one hour; and continues in the same fashion, increasing the number of candles lighted each Sunday by one, until in the $n$th Sunday, she lights up all the $n$ candles for one hour. For what values of $n$ is it possible for all the $n$ candles to be of equal lengths right after the $n$th Sunday?
60.5. all positive odd integers
Since the candles are identical and they are of equal lengths right after the $n$th Sunday, they are used equal number of times for all the $n$ Sundays. The total number of times they are used for the $n$ Sundays is
$$
1+2+3+\cdots+n=\frac{n(n+1)}{2} .
$$

It follows that each candle is used $\frac{1}{2}(n+1)$ times for all the Sundays. Thus, since $\frac{1}{2}(n+1)$ must be an integer, $n$ must be odd.

We need to exhibit a procedure to show that, when $n$ is odd, it is indeed possible for Sharon to carry out the lighting of the candles. We first label the candles with the numbers $1,2, \ldots, n$. We arrange these numbers in triangular array, writing $1,2, \ldots, n$ consecutively, and goes back to 1 after $n$, until we complete the $n$ Sundays. The procedure is illustrated below when $n=7$ :