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Problem of the Week
Problem B
Crazy Grids

The grid below consists of four rows, each with five 1 by 1 squares. Seven of these squares contain groups of dogs. These dogs play nicely with other dogs in their own group, but do not play well with the dogs in other groups. You would like to separate the groups of dogs by putting up fences along some of the dashed gridlines.

All gridlines are dashed lines. The seven squares that have a group of dogs and the number of dogs in each of these squares is provided in the adjacent list.

  1. Find two ways to put up fences that would divide the grid into seven distinct regions, so that each region contains the same number of 1 by 1 squares as dogs.
    An example of a different 3 by 3 grid divided into three such regions is shown in the diagram.

Vertical and horizontal line segments are drawn along some gridlines making regions. One region has two squares side by side and contains 2 dogs in total. Another region has three squares edge to edge forming a capital L shape, and contains 3 dogs. The final region has four squares edge to edge forming a capital L shape and contains 4 dogs.

  1. Share your solutions with your classmates. Did they find solutions different from yours?

  2. Move the seven groups of dogs to different squares in the grid, and repeat part a) using your new grid. Did you find any arrangements for which there was NO solution?

Not printing this page? You can divide the grid on our interactive worksheet.