Knights and liars in island

Question

There are liars and knights in some island (liars always lie and knights always say true). $500$ people were built in the form of a rectangle $20\times 25$ ($20$ people in column and $25$ in row). During the poll, everyone stated: 1) If you do not count me, there are more knights in my column than liars. 2) If you do not count me, there more liars in my row than knights. Estimate the number of knights in rectangle.

I have spent some time in order to find some approach but no results. Can anyone explain the solution of this problem.

Are (1) and (2) distinct depositions, so that a knave is supposed to lie in both of them? — Fabio Somenzi, Jan 27 '18 at 21:23
The original terminology, by the way, is “knights and knaves”. Is this out of one of Smullyan’s books? — Alexander Burstein, Jan 27 '18 at 21:30

score 3 · Accepted Answer · answered Jan 27 '18 at 20:39

Assume that in a column there are $k$ knights and $20-k$ liars. If $k\geq 1$, it must be true that $k-1>20-k$, hence $k\geq 11$. If in such a column there is a liar, it must be true that $k\leq 19-k$, hence in any column there cannot be a liar and a knight at the same time: each column is either made by knights only or by liars only.

Assume that in a row there are $k$ knights and $25-k$ liars. If $k\geq 1$, it must be true that $25-k>k-1$, hence $k\leq 12$. In particular a row cannot be made by knights only, and in each row there is a liar. This gives us that $24-k\leq k$, hence $k\geq 12$.

Conclusion: there are $12$ columns of knights and $13$ columns of liars.

Very nice solution! However, I was too close but my approach was false :( — RFZ, Jan 27 '18 at 20:51

Knights and liars in island

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