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?

KLLKLLK
LKLLKLL
LLKLLKL
KLLKLLK
LKLLKLL
LLKLLKL

This pattern meets the criteria that all can say that they are standing next to people of the opposite type - all the knights are by liars and all the liars are by a mixture of knights and liars - but only 33% are knights, so I don't know how you could add a few more knights in.

Posted by Lewis on 2003-06-24 10:58:45