On the picture, you can see an example of a wall made of 2×1 bricks. On the wall, there are 2 cracks, which are straight lines passing through the whole wall from top to bottom and from left to right, without intersecting any bricks.
Can you make the following walls without any cracks:
- wall 5×6 with 15 bricks;
- wall 6×6 with 18 bricks?
The solution for a 5×6 wall is shown below. However, if the wall has dimensions 6×6, it is impossible to build it without any cracks. Indeed, assume the wall does not have any cracks. Therefore every line passing through it must intersect 2, 4, or 6 bricks. Since there are in total 10 lines passing through the wall and each brick is intersected by exactly one of them, the total number of bricks must be at least 10 x 2 = 20 > 18. This yields a contradiction.