town of $P$ which he was commanding. Escaping, $H$ found himself trapped in a cave with $40$ companions. The Romans discovered his whereabouts and

$3$

invited him to surrender, but his companions refused to allow him to do so$.$ He therefore suggested that they kill each other, one by one, the order to be decided by lot. Tradition has it that the means for effecting the lot was to stand in a circle, and, beginning at some point, count round, every third person being killed in turn. The sole survivor of this process was $H,$ who then surrendered to the Romans. Which begs the question: had $H$ previously practiced quietly with $41$ stones in a dark corner, or had he calculated mathematically that he should adopt the $31$st position in order to survive?

Having read an account of this gruesome event you become obsessed with the fear that you will find yourself in a similar situation at some time in the future. In order to prepare yourself for such an eventuality you decide to write a program to run on your mobile phone which will determine the position in which you should stand to ensure that you will be the sole survivor.

In particular, your program should be able to handle the following processes. $n>0$ people are initially arranged in a circle, facing inwards, and numbered from $1$ to $n$ clockwise. The numbering from $1$ to $n$ proceeds consecutively in a clockwise direction. Starting with person number $1,$ counting starts in a clockwise direction, until we get to person number $)>(0,$ who is promptly killed. We then proceed to count a further $k$ people in a clockwise direction, starting with the person immediately to the left of the victim $($that is$,$ the victim's left$).$ That kth