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NIce :) Here is a solution that shows only who wins, not how Alice wins. As the game is finite, some player has to have winning strategy, suppose it is Bob. As a part of the strategy, Bob must erase k (with all its factors) when Alice's first move is erasing 1. But now, Alice simply chooses k at the beginning, stealing Bob's winning strategy. This shows Bob can't win, so Alice must have a winning strategy.
who’s to say that if Alice chooses k that this doesn’t change the state of the game such that Bob can choose winning move j? The reasoning sounds circular: “assume there is a winning first move for Bob. Alice plays this move and wins”
@@shyampadmanabhan4171 We do not care what Bob plays after that k, imagine Bob has the piece of paper that says what exacly he does in every situation, i.e. his winning strategy. The main observation is that erasing number 1 does virtually nothing. He can then share the paper with Alice as this will not affect the outcome of the game (assuming Bob is guaranteed to win). Alice then looks what is Bob's move after erasing number 1 and erases that number instead (and of course number 1 in the process). Existence of Bob's strategy would imply that Bob now has the winning strategy both in the case it is his or Alice's turn, which is nonsense. Hope this clarifies things a bit
@@bruhmomenat4598 What if Bob has two winning strategies? Alice can only wipe out one. I don’t think we have grounds to assume how many, if any, winning strategies Bob has
@@shyampadmanabhan4171 it is not important. The main thing is that in any finite two player game, one cannot guarantee a win both starting first and starting second (intuitively this should make sense). This is the case here; if we assume Bob has the winning strategy starting second, then he also has to have a winning strategy starting first (with adapted ), not possible due to the above paragraph
There are already many resources for programming problems. Can you suggest something the current resources lack? One thing I can think of is that they don’t focus much on how to arrive at the solution or how to develop intuition.
@@Quant_ProfI bought your course and it's really amazing - covers most of the math I would need for my interviews. However there are no resources available online for company wise programming questions (leetcode mainly has MAANG and barely any quant companies) so I would really appreciate it if you could add links to company wise programming questions in the course (the company playlists for math questions are really helpful)
Answer for follow up: Suppose that Alice has a winning strategy without removing only 1. If she does, she'll win. If she doesn't, then by removing 1, she gives the identical board (since any move removes 1) to Bob, who then must lose, meaning Alice wins. Alice wins no matter what.
🎓 Ready to level up and gain a significant edge? Check out the Quant Interview Masterclass for an in-depth guide to mastering quant interviews (with *LIFETIME ACCESS* and *30 days FULL REFUND* policy):
payhip.com/b/heLaQ
NIce :) Here is a solution that shows only who wins, not how Alice wins. As the game is finite, some player has to have winning strategy, suppose it is Bob. As a part of the strategy, Bob must erase k (with all its factors) when Alice's first move is erasing 1. But now, Alice simply chooses k at the beginning, stealing Bob's winning strategy. This shows Bob can't win, so Alice must have a winning strategy.
correct, you got it.
this method is known as "stealing the strategy".
who’s to say that if Alice chooses k that this doesn’t change the state of the game such that Bob can choose winning move j? The reasoning sounds circular: “assume there is a winning first move for Bob. Alice plays this move and wins”
@@shyampadmanabhan4171 We do not care what Bob plays after that k, imagine Bob has the piece of paper that says what exacly he does in every situation, i.e. his winning strategy. The main observation is that erasing number 1 does virtually nothing. He can then share the paper with Alice as this will not affect the outcome of the game (assuming Bob is guaranteed to win). Alice then looks what is Bob's move after erasing number 1 and erases that number instead (and of course number 1 in the process). Existence of Bob's strategy would imply that Bob now has the winning strategy both in the case it is his or Alice's turn, which is nonsense. Hope this clarifies things a bit
@@bruhmomenat4598 What if Bob has two winning strategies? Alice can only wipe out one. I don’t think we have grounds to assume how many, if any, winning strategies Bob has
@@shyampadmanabhan4171 it is not important. The main thing is that in any finite two player game, one cannot guarantee a win both starting first and starting second (intuitively this should make sense).
This is the case here; if we assume Bob has the winning strategy starting second, then he also has to have a winning strategy starting first (with adapted ), not possible due to the above paragraph
can you upload programming problems (data structures, algorithms related problems asked in quant) as well please?
btw love your content, straight forward and to the point
There are already many resources for programming problems. Can you suggest something the current resources lack? One thing I can think of is that they don’t focus much on how to arrive at the solution or how to develop intuition.
@@Quant_ProfI bought your course and it's really amazing - covers most of the math I would need for my interviews. However there are no resources available online for company wise programming questions (leetcode mainly has MAANG and barely any quant companies) so I would really appreciate it if you could add links to company wise programming questions in the course (the company playlists for math questions are really helpful)
hacker rank, leet code
dude just go on hacker rank on do those red questions, the red ones are HARD
Answer for follow up: Suppose that Alice has a winning strategy without removing only 1. If she does, she'll win. If she doesn't, then by removing 1, she gives the identical board (since any move removes 1) to Bob, who then must lose, meaning Alice wins. Alice wins no matter what.