# Problem 3

Let \(n\) be a positive integer. Liu Bang and Xiang Yu have a stick of length 1 and want to divide it between themselves. Liu first marks at most \(n\) points on the stick, and then Xiang marks at most \(n\) points on the stick. All marked points are distinct. The stick is then cut at every marked point, producing a collection of pieces.

Afterwards, Liu and Xiang take turns claiming any unclaimed piece, with Liu moving first. Each player wants to maximise the total length of the pieces they receive.

For each \(n\), determine the largest number \(c\) such that Liu can guarantee a total length of at least \(c\), regardless of Xiang's play.