There might be cases when the number of coins to pick in first turn could come up to be zero (from the formula). Consider the following case.
QUESTION: Two smart players A and B are playing a coin game in which they can pick up 1, 2, 3 or 4 coins. They have 76 coins and the player who picks the last coin will lose the game. A and B play alternately and A plays the first move. How many coins should A pick at first so his win is independent of number of coins B picks in his first move?
On solving with the methods known, the answer for the number of coins A should pick at first will come out to be zero. But since he has to pick up at least 1 coin, in that case the control factor will shift into the hands of B. Practically, in such a situation A cannot win the game no matter how many coins he picked up in his first move as the control factor will shift to B and since B is also a smart player, he will keep the control factor in his hands and win the game.
WHEN THE REMAINDER IS LESS THAN THE MINIMUN NUMBER OF COINS:
In addition to the remainder coming out to be zero, there can be cases where the remainder comes out to be less than the minimum number of coins allowed. Let’s take an example to understand.
QUESTION: Two smart players A and B are playing a coin game in which they can pick up 2, 3 or 4 coins. They have 75 coins and the player who picks the last coin will lose the game. A and B play alternately and A plays the first move. How many coins should A pick at first so his win is independent of number of coins B picks in his first move?
Here, when we try to find the remainder so as to get the number of coins for the first turn, it will come to be 1 but picking 1 coin is not allowed, A has to pick at least 2 coins in his first turn. These type of cases might end in a draw otherwise provided that even if the number of coins remaining for the last turn is less than the minimum but still the last person has to pick them up and he loses. If nothing of such sort is mentioned then to draw the game A would want to leave 1 coin for the last. The control factor will then be (min + max)*k + 1 and so the minimum number of coins A will pick up to maintain this control factor will be 2 as per
A very general formula which will help in all kinds of coin problems can be expressed as:
Where, ‘coins to be left for last turn’ can be changed as per the given situation of the question. We’ve already derived the same for some cases but logically thinking about the number of coins one would want to leave to win the game for other person’s last turn will give us the number of coins he should pick up at the start. In some questions instead of coins, matchsticks can be used but still the approach remains the same.