Dua has all the odd natural numbers less than 20. Asija has all the even numbers less than 21. They play the following game. In each round, they take a number from each other and after every round, they may fix two or more consecutive numbers so that their opponent cannot take these fixed numbers in the next round. The game is won by the player who attains 10 consecutive numbers first. Does either player have a winning strategy?
Problem
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L567
28.02.2021 13:26
I don't get this problem... if you can fix two or more consecutive numbers... why not fix all the numbers so that the opponent cant take any number from you next round and so noone wins? @below - Thanks, got it
bsf714
28.02.2021 14:24
L567 wrote: I don't get this problem... if you can fix two or more consecutive numbers... why not fix all the numbers so that the opponent cant take any number from you next round and so noone wins? Note that Dua has odd numbers whereas Asija has even ones. They cannot fix everything immediately as they do not possess consecutive numbers in the beginning.
B11
14.03.2021 02:06
They both have 20 numbers. If one of them made 10 consecutive numbers, the other one would have 10 consecutive numbers too. So neither of them can't win the game
EdiBerisha
28.03.2021 16:09
B11 wrote: They both have 20 numbers. If one of them made 10 consecutive numbers, the other one would have 10 consecutive numbers too. So neither of them can't win the game Notice how the problem, specifically tells us: "The game is won by the player who attains 10 consecutive numbers FIRST." Now, how can someone win? Just as an example: Let's say both players tie at 9 consecutive numbers, it is clear that the one who plays next will get 10 consecutive numbers, while the other needs one more move to attain 10. Now with that logic, let's say at the beginning they're 2-2 with consecutive numbers, then 3-3, and so on... whoever played first from the beginning, will have the chance to get 10 consecutive numbers first, when the score is 9-9. Meaning that the answer is, the player that plays first wins. But is that all? How does the first player, make the situation similar as the example? This could be done if that player picks the number 10. Why? Because that is the median of the number line from 1 to 21. And it is important to pick this number, because if the second player decides to take a number and fix just to block the first player from winning, it'll be a disadvantage for the second player because the first player has still the ability to win, by going the opposite side of the latter's targeted.