Building with Bricks

In this puzzle we take on more mathematical functions.

An architect is building a series of pyramids to use in landscaping a fancy garden. The pyramids are going to be of different heights, and she wants to know how many bricks she will need to build a pyramid of any height.

The pyramids are built using a very specific rule. Every block must be exactly on top of another block. This is what the pyramids look like: bricks

How many bricks will the architect need to build a pyramid five rows high? Ten rows high? Can you find a rule that will tell her the number of bricks, B, she will need to build any pyramid of height, H.

Background

In the Seats at the Table or “No Function Too Large” [1] math puzzle, we deal with functions. Remember that in mathematicsspeak a function is a rule relating two variables, so that if you know one of the variables, called the independent variable, the rule tells you the value of the other variable, the dependent variable.

Hint: This is a different situation than last month, with a different kind of function; however, the basic strategies for solving the puzzle are the same as for last month. Make drawings on graph paper, or use blocks to build structures for the puzzle. Keep track of your results in a table listing the variables. Make a conjecture about a rule that will give the same results as the data in your table.

Course:

Math [2]
Algebra [3]
Geometry [4]

Result/Solution(s)

Solution: Building With Bricks Math Puzzle

There are a number of different ways that students have solved this puzzle.

Marisa's Solution

I found some sugar cubes and used them to build several pyramids. I had only about 30 cubes, so I had to keep reusing the same cubes. After a few tries I found a shortcut. After I built one pyramid, I’d build the next base, then pick up the previous pyramid and place it on the base. As I was doing this, I noticed that I had to add 2 more bricks for each new base than for the base of the previous pyramid.

I kept track of my results in a table:

H = height of pyramid	B = number of bricks	A = bricks added
1	1	--
2	4	3
3	9	5
4	16	7
5	25	9

I noticed that the number of bricks kept increasing by 2, so I knew that after a pyramid of height 5, I had to add 11 bricks for a pyramid of height 7, add 13 to that for a pyramid of height 7, and so forth, so I worked my way up to 10. I didn’t build any more pyramids because I was out of cubes.

H = height of pyramid	B = number of bricks	A = bricks added
1	1	--
2	4	3
3	9	5
4	16	7
5	25	9
6	36	11
7	49	13
8	64	15
9	81	17
10	100	19

So my rule is: If you know how many bricks you used for one pyramid, you have to add 2 more bricks than you added the last time, to get the next pyramid.

Sampen's Solution

I used graph paper to work out my solution. I made drawings of five pyramids.

Then I made a table.

H	B
1	1
2	4
3	9
4	16
5	25

I could see from the table that B is always H x H or B = H2. Then I drew another pyramid 6 bricks high. It needed 36 bricks, which is 6 x 6.

So my rule is correct.

Andre's Solution

I used square tiles to build my pyramids. I noticed that after I built a pyramid, I could always rearrange the tiles into a square. What I did was take away all the tiles to the right of the middle stack and move them all over to the left. They always fit in and make a square. The height of the square is the same as the height of the pyramid. So the number of square tiles is the same as the area of the square or H x H. The rule for the architect is B = H x H.

Here are some examples:

H = 2, area of big square = 2 x 2 = 4

H = 3; area of big square = 3 x 3 = 9

H = 6; area of big square = 6 x 6 = 36