Download Skeleton Tower How many cubes are needed to build this tower

Skeleton Tower
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How many cubes are needed to build this tower?
How many cubes are needed to build a tower like this,
but 12 cubes high?
Explain how you worked out your answer to the first
question.
How would you calculate the number of cubes needed
for a tower n cubes high?
How many cubes are needed to build this tower?
66 cubes are needed to build the Skeleton tower that is 6 cubes high.
How many cubes are needed to build a tower like this, but 12 cubes high?
276 cubes are needed to build the Skeleton tower that is 12 cubes high.
Explain how you worked out your answer to the first question.
Since 6 cubes high is countable, I counted how many cubes are needed to build the tower. Although I
was able to count the cubes, I also thought of how I would figure this out on a large scale. The first
method I chose to try is dividing the tower into four sections. In each section there are 15 cubes, but
there are also 6 cubes that runs through the center of the tower. So, 15(4)+6 = 66 cubes. But I thought
this could become difficult when the number of cubes gets higher.
Instead of breaking up the tower into four
sections, I broke it into two half sections like
demonstrated. Each level of cubes was an odd
number, so I had to figure out a method to sum
up the odd numbers. I created a summation
equation
that will
count the number of cubes. The summation
counts the odd numbers, starting at one and
going up to x cubes high. Since there are two
halves, like the diagram, the summation must
be multiplied by two. Two times the summation is then subtracted by x because the middle stack of
cubes is counted twice, and x counts the height of the tower. But once I figured out this formula I ran
into another problem. As x gets larger, the summation will become too lengthy. I remembered a story
about a famous mathematician who summed up the digits from 1 to 100 at a very young age with an
incredible method. Using his idea by arranging the numbers
1
2
3
4 … 100
100
99
98
97 … 1
101
101
101
101 … 101
101(100) 2 = 5050
Taking this same idea, I replaced my summation formula with
where the
counting property is within the [ ] and x is the number of cubes high that the tower is.
6 cubes:
= 2[(2(5) + 2)(6) 2 ] – 6
= 2[(12)(6) 2 ] – 6
= 2(36) – 6
= 72-6
= 66
12 cubes:
= 2[(2(11) + 2)(12) 2 ] – 12
= 2[(24)(12) 2 ] – 12
= 2(144)-12
=288-12
= 276
How would you calculate the number of cubes needed for a tower n cubes high?
Using the same formula, the number of cubes needed for a tower n cubes can be found with
.