Given that there are 21 flags, NOW using backward induction we can see that;
If there is only one flag left, whosoever picks up that particular flag will lose. Therefore, 1 is loosing number.
Now if there are 2 flags left, whosoever's turn it is, can pick 1 flag and leave one for the opponent and win the game. Therefore, 2 is winning number. Similarly if there are 3 flags left then whosoever's turn it is can pick 2 flags and leave one for the opponent and can win the game. Hence 3 is winning number. Similarly when 4 flags left, 3 can be picked up and leave one for the opponent and win the game. So, 4 is also winning number.
Now, if there are 5 flags left, then no matter whichever number you choose among 1,2 or 3 you gonna leave other team with 4,3 or 2 respectively and all 3 are winning number. Therefore, 5 is loosing number.
Similarly, if you have 6,7,8 then by taking 1,2,3 respectively you can leaveyour opponent with number 5 and that is loosing number and you can win certainly. Therefore, 6,7,8 are also winning number.
Following above discussion, we see a pattern among the number that would make sure you win and with other numbers you loose.
we see after 1 if we go in positive direction on number line till 21 then every fourth number is loosing number because no matter what you choose you will leave your opponent with a winning number of flags and if your opponent knows and follows the optimal strategy can win. Therefore,
Loosing Numbers = 1,5,9,13,17 and 21
Winning Numbers = 2,3,4,6,7,8,10,11,12,14,15,16,18,19 and 20
a) STRATEGY
If you are a team that is stuck with any of loosing number that is 1,5,9,13,17 or 21 then no matter what strategy you follow you gonna loose. Therefore, there is n winning strategy in this situation given other team also knows the winning strategy
But if you have any of winning numbers at any point then you can follow the following strategy and can win
- 2 flags = pick up one flag and leave 1 for the opponent, win
- 3 flags = pick up 2 flags and leave 1 for the opponent, win
- 4 flags = pick up 3 flags and leave 1 for the opponent, win
- 6 flags = pick up one flag, then no matter what your opponent chooses you follow either strategy number 1,2 or 3 based on the number of flags left and win
- 7 flags = pick up 2 flags, then no matter what your opponent chooses you follow either strategy number 1,2 or 3 based on the number of flags left and win
- 8 flags = pick up 3 flags, then no matter what your opponent chooses you follow either strategy number 1,2 or 3 based on the number of flags left and win
- 10 flags = pick up one flag, then no matter what your opponent chooses you follow either strategy number 4,5 or 6 based on the number of flags left and win
- 11 flags = pick up 2 flags, then no matter what your opponent chooses you follow either strategy number 4,5 or 6 based on the number of flags left and win
- 12 flags = pick up 3 flags, then no matter what your opponent chooses you follow either strategy number 4,5 or 6 based on the number of flags left and win
- 14 flags = pick up one flag, then no matter what your opponent chooses you follow either strategy number 7,8 or 9 based on the number of flags left and win
- 15 flags = pick up 2 flags, then no matter what your opponent chooses you follow either strategy number 7,8 or 9 based on the number of flags left and win
- 16 flags = pick up 3 flags, then no matter what your opponent chooses you follow either strategy number 7,8 or 9 based on the number of flags left and win
- 18 flags = pick up one flag, then no matter what your opponent chooses you follow either strategy number 10,11 or 12 based on the number of flags left and win
- 19 flags = pick up 2 flags, then no matter what your opponent chooses you follow either strategy number 10,11 or 12 based on the number of flags left and win
- 20 flags = pick up 3 flags, then no matter what your opponent chooses you follow either strategy number 10,11 or 12 based on the number of flags left and win
b)
Since team moving first will have 21 flags left on ground and it is a loosing number, no matter which number it chooses to pick it is gonna loose as it will leave other team with winning number. So, they will be indifferent between choosing 1,2 or 3
c)
If first teams removes 1 flag then best strategy for second team would be to remove 3 flags and again leave first team with loosing number
similarly, If first teams removes 2 flags then best strategy for second team would be to remove 2 flags and again leave first team with loosing number
and If first teams removes 3 flags then best strategy for second team would be to remove only 1 flag and again leave first team with loosing number
