The Skeleton Tower

You remember the task I had to solve about chicken nuggets? Well, this week, I had to explore another task as a graduate student.

Note the figure below. It is made of cubes that form stair steps. The center of the tower is 6 cubes high. On all four sides of the top tower are stair steps that extend from the center.

How many cubes are needed to build a tower like this, but that is 3 cubes high? 4 cubes high? 5 cubes high? N cubes high?

With the chicken nugget task, it took me a while to see the pattern to solve the problem. With this task, however, the connection between the number of blocks in the center, and the number of blocks in each of the 4 sides can be seen pretty quickly.

• To build a 3 cubes high tower, you would need: 3 + 4 x (2+1) = 15 blocks
• To build a 4 cubes high tower, you would need : 4 + 4 x (3+2+1) = 28 blocks
• To build a 5 cubes high tower, you would need: 5 + 4 x (4+3+2+1) = 45 blocks
• To build a n cubes high tower, you would need: n + 4 x ((n-1) + (n-2) + (n-3) + …. + 1)

A chart, a table, may help you figure out the pattern, but a model was the approach I chose.