LEGO R
bricks are very common toys and have a structure that allows them to be placed
on a rectangular grid. This makes them great for interesting questions about math. A 2�4
brick has two rows and four columns, making it have 8 total studs. But a 2�4 is the same
as a 4 � 2, just rotated 90 degrees.
In this assignment, you are going to �nd how many ways you can put a 2 � 4 on top
of another 2 � 4. We will give some simple examples of what this means. If we have a
1�1, there are 8 places on the top of a 2�4 we can place it if we don't move or rotate the
bottom. But, some of these positions are going to be the same. If I place a 1�1 in the top
left corner of a 2 � 4, and rotate the 2 � 4 by 180�, we see that it is the same as placing a
1 � 1 in the bottom right of the 2 � 4.
So, there are only four ways to put a 1 � 1 on a 2 � 4, since placing a 1 � 1 in any spot
on the top row will be moved to a spot on the bottom row by rotating the whole brick. If
we used a 1 � 2 instead of a 1 � 1, we would have more ways, since the brick can hang o�
the side. Picture below are the only two ways to place a 1 � 2 on the top left stud.
For this assignment, you will:
(1) Find the number of positions you can place a 1�2 on top of a 2�4 without moving
or rotating the bottom 2 � 4. (You can rotate the 1 � 2, but only by 90�.) Be sure
justify why you found every way and there are no others.
(2) Use part (1) to �nd out how many ways you can place a 1 � 2 on top of a 2 � 4 if
you are now allowed to rotate the bottom brick after you place the top brick. Be
sure justify why you found every way and there are no others.
(3) Instead of using a 1 � 2 to put on top of a 2 � 4, use a 2 � 4 to:
(a) Repeat part (1). Be sure to justify your solution as you did in part (1).
(b) Repeat part (2). Be sure to justify your solution as you did in part (2).