24. (IND 1) An odd integer \( n \geq 3 \) is said to be "nice" if there is at least one permutation \( a_{1}, a_{2}, \ldots, a_{n} \) of \( 1, 2, \ldots, n \) such that the \( n \) sums \( a_{1}-a_{2}+a_{3}-\cdots-a_{n-1}+a_{n}, a_{2}-a_{3}+a_{4}-\cdots-a_{n}+a_{1}, a_{3}-a_{4}+a_{5}-\cdots-a_{1}+a_{2}, \ldots, a_{n}-a_{1}+a_{2}-\cdots-a_{n-2}+a_{n-1} \) are all positive. Determine the set of all "nice" integers.
24. Let \( y_{k} = a_{k} - a_{k+1} + a_{k+2} - \cdots + a_{k+n-1} \) for \( k = 1, 2, \ldots, n \), where we define \( x_{i+n} = x_{i} \) for \( 1 \leq i \leq n \). We then have \( y_{1} + y_{2} = 2 a_{1}, y_{2} + y_{3} = 2 a_{2}, \ldots, y_{n} + y_{1} = 2 a_{n} \). (i) Let \( n = 4k - 1 \) for some integer \( k > 0 \). Then for each \( i = 1, 2, \ldots, n \) we have that \( y_{i} = (a_{i} + a_{i+1} + \cdots + a_{i-1}) - 2(a_{i+1} + a_{i+3} + \cdots + a_{i-2}) = 1 + 2 + \cdots + (4k - 1) - 2(a_{i+1} + a_{i+3} + \cdots + a_{i-2}) \) is even. Suppose now that \( a_{1}, \ldots, a_{n} \) is a good permutation. Then each \( y_{i} \) is positive and even, so \( y_{i} \geq 2 \). But for some \( t \in \{1, \ldots, n\} \) we must have \( a_{t} = 1 \), and thus \( y_{t} + y_{t+1} = 2 a_{t} = 2 \) which is impossible. Hence the numbers \( n = 4k - 1 \) are not good. (ii) Let \( n = 4k + 1 \) for some integer \( k > 0 \). Then \( 2, 4, \ldots, 4k, 4k + 1, 4k - 1, \ldots, 3, 1 \) is a permutation with the desired property. Indeed, in this case \( y_{1} = y_{4k+1} = 1, y_{2} = y_{4k} = 3, \ldots, y_{2k} = y_{2k+2} = 4k - 1 \), \( y_{2k+1} = 4k + 1 \). Therefore all nice numbers are given by \( 4k + 1, k \in \mathbb{N} \).