SinceWe are supposed to maximize the least number of marbles he could have picked up, let us begin by assuming first that the least possible number of marbles he could have picked up in first two rounds was 201, (1st round: 101 and 2nd round: 100).Now, in the next three rounds he would have picked up 136 marbles (337 - 101). To maximize the least number of marbles, the number of marbles he might have picked up in 3rd, 4th and 5th round should be as close as possible.This gives us possible breakup of 47, 45 and 44 marbles respectively.Now, from round 6 the number of marbles that he could have picked up can be in decreasing order (decremented by 1 at each step) as shown.The least number of marbles that he could have picked up in any of the rounds can be at most 24.Answer: 24