문제<\/h2>
Juraj Jánošík (1688–1713) was a legendary Slovak outlaw, said to take from the rich and give to the poor. According to our sources, he did so on a countryside of a grid shape. The grid consists of $(<\/mo>N<\/mi>+<\/mo>1<\/mn>)<\/mo>\times<\/mo>(<\/mo>M<\/mi>+<\/mo>1<\/mn>)<\/mo><\/mrow>(N + 1) \times (M + 1)<\/math><\/span><\/span> (<\/span> N<\/span> <\/span> +<\/span> <\/span><\/span> <\/span> 1<\/span>)<\/span> <\/span> \times<\/span> <\/span><\/span> <\/span> (<\/span> M<\/span> <\/span> +<\/span> <\/span><\/span> <\/span> 1<\/span>)<\/span><\/span><\/span><\/span> intersections, with each pair of neighboring intersections connected by a road. Along each road, there is a house, which is either rich or poor. Each region enclosed by 4<\/mn><\/mrow>4<\/math><\/span><\/span> 4<\/span><\/span><\/span><\/span> roads is called a field.<\/p> Now, Jánošík works in a very particular way. Each day, he picks a path along the roads and visits all houses on those roads. If the house is rich, then he of course takes from them, making them poor. On the other hand, if the house is poor, he will give to them, making them rich. Jánošík carries with him enough money, to make any number of houses rich. Similarly, Jánošík has big enough pockets to steal from any number of houses.<\/p> Furthermore, the paths chosen by Jánošík always have a particular property. Jánošík<\/p> either drives in a straight line, going from one border of the country to the opposite one, visiting each house exactly once along this straight line,<\/li> or he picks a connected area of fields, and walks along the roads on the perimeter of this area, visiting each house exactly once.<\/li>
<\/ul> Recently, the historians found records, showing which houses were rich and which were poor in year 1687, and another record from year 1714, both from the same country. They want you to find out whether the redistribution of wealth can be only due to Jánošík. Jánošík might have performed his redistributions of wealth$

JJ the Almighty