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The problem is named after Flavius Josephus, a Jewish historian and leader who lived in the 1st century. According to Josephus's firsthand account of the siege of Yodfat, he and his 40 soldiers were trapped in a cave by Roman soldiers. They chose suicide over capture, and settled on a serial method of committing suicide by drawing lots.
Then it gives a proof that uses a different specific example ("we explicitly solve the problem when every second person will be killed"). But in no case does it actually give the answer. I believe the answer for the Josephus example of 41 participants and a step of three is that position 31 is the survivor and position 16 is the next-to-last.
The problem for graphs is NP-complete if the edge lengths are assumed integers. The problem for points on the plane is NP-complete with the discretized Euclidean metric and rectilinear metric. The problem is known to be NP-hard with the (non-discretized) Euclidean metric. [3]: ND22, ND23
Josephus problem table: Image title: Claude Gaspar Bachet de Méziriac's interpretation of the Josephus problem with 41 soldiers and a step size of 3, visualised by CMG Lee. Time progresses inwards along the spiral, green dots denoting live soldiers, grey dead soldiers, and crosses killings.
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Hilbert's tenth problem: the problem of deciding whether a Diophantine equation (multivariable polynomial equation) has a solution in integers. Determining whether a given initial point with rational coordinates is periodic, or whether it lies in the basin of attraction of a given open set, in a piecewise-linear iterated map in two dimensions ...
Josephus problem 30 9: Image title: Variant of the Josephus problem with 15 Christians and 15 Turks and a step size of 9, visualised by CMG Lee. Time progresses inwards along the spiral, green dots denoting live soldiers, grey dead soldiers, and crosses killings. Width: 100%: Height: 100%
Although Josephus says that he describes the events contained in Antiquities "in the order of time that belongs to them," [63] Feldman argues that Josephus "aimed to organize [his] material systematically rather than chronologically" and had a scope that "ranged far beyond mere political history to political institutions, religious and private ...