Comments on: two inductive problems https://blog.yhuang.org/?p=800 here. Tue, 14 Oct 2025 11:10:14 +0000 hourly 1 http://wordpress.org/?v=3.1.1 By: me https://blog.yhuang.org/?p=800&cpage=1#comment-113345 me Sat, 03 Nov 2012 19:30:54 +0000 http://scripts.mit.edu/~zong/wpress/?p=800#comment-113345 The induction is not "This is true for day number K" or "This is true for the number of blue-eyes K." It's "This K-indexed inference about the relationship between day number and the number of blue-eyes is true for K" where K is just an abstract counter. Inside the inference we can say anything we want. But maybe I'm not understanding your comment, since even if there were some induction on the number of blue-eyes K directly, you would still be proving it from 0 onward so what is this about the number of islanders not changing? The induction is not “This is true for day number K” or “This is true for the number of blue-eyes K.” It’s “This K-indexed inference about the relationship between day number and the number of blue-eyes is true for K” where K is just an abstract counter. Inside the inference we can say anything we want.

But maybe I’m not understanding your comment, since even if there were some induction on the number of blue-eyes K directly, you would still be proving it from 0 onward so what is this about the number of islanders not changing?

]]> By: Jack https://blog.yhuang.org/?p=800&cpage=1#comment-113344 Jack Sat, 03 Nov 2012 10:45:14 +0000 http://scripts.mit.edu/~zong/wpress/?p=800#comment-113344 Your proof is using k for two different things, the day number and the number of blue-eyed islanders. You can do induction on the day, but not on the number of islanders because that does not change during the problem. So while the theorem is true, your proof is not valid. Your proof is using k for two different things, the day number and the number of blue-eyed islanders. You can do induction on the day, but not on the number of islanders because that does not change during the problem. So while the theorem is true, your proof is not valid.

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