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I am having trouble solving the following problem:

An expedition of n people has discovered a treasure in the mountains; each pair of them can carry out one piece. A coalitional game that models this situation is $(N,v)$, where $v(S) = |S|/2$ if $|S|$ is even and $v(S) = (|S| − 1)/2$ if $|S|$ is odd. Find the core and the Shapley value of this game.

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This is Example 259.2 in Osborne (1994), A Course in Game Theory.

"If $|N| \ge 4$ is even, then the core consists of the single payoff profile $(1/1, \ldots, 1/2)$. If $|N|\ge 3$ is odd, then the core is empty."

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