Kazaam the wizard lives in the kingdom of Liars and Knights. He is planning a grand illusion for the King's golden jubilee, in which he will make hundreds of people appear to turn into gold.

Kazaam plans to lay out markers on the parade ground for the people in the illusion to stand on. For the illusion to work, the markers must be laid out in a perfectly square grid, with an even number of rows and columns. Every marker must have a liar or knight standing on it, arranged such that they each can say that every person standing next to them in the same row or the same column is of the opposite persuasion (i.e. every knight can say that all adjacent markers have a liar standing on it, and vice versa).

The last detail required for Kazaam's spell to work is that at least 37% of the people in the illusion must be knights. What is the minimum number of rows and columns needed to accommodate this ratio of knights to liars, keeping in mind Kazaam wants at least 100 people in the illusion?

(In reply to Am I missing something by K)

If you have an alternating grid, such as the red and black squares on a checkerboard, each knight is surrounded by all liars (which we want), and each liars is surrounded by all knights. However, being a liar, he will not say that he is surrounded by all knights, and the arrangement fails.

To solve this problem, we need to find an arrangement for which each knight is surrounded by all liars, but each liar is next to at least one other liar (so he can lie and say that all his neighbors are knights).

Posted by DJ on 2003-06-24 13:08:09