Summing squares
Discover a way to sum square numbers by building cuboids from small
cubes. Can you picture how the sequence will grow?
Age
14 to 16
Challenge level
Problem
You may wish to make six copies of each square prism below to aid you in visualising the problem that follows.
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If I take six of the
red cubes I can make this 3 by 2 by 1 cuboid:
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I want to surround this with blue
cubes to make the 5 by 3 by 2 cuboid below:
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Could I build this cuboid by adding on six blue square prisms like the
one above without splitting any of them?
If so, describe how to do it. If not, explain why not, and
describe how to arrange the blue square prisms so that as few as
possible need to be split.
I want to surround this with pink
cubes to make a 7 by 4 by 3 cuboid. (See Hint for a picture of this
cuboid.)Could I build it by adding on six pink square prisms like the
one above? Do I need to split any this time?
So far, we've discovered that
$$6 \times 1^2 = 3 \times2 \times1$$
$$6\times(1^2+2^2) = 5\times3\times2$$
$$6\times(1^2+2^2+3^2) = 7\times4\times3$$
Can you describe a method for forming the next cuboid in the
sequence by adding on six
square prisms? You will only need to split one of your square
prisms each time. Does the sequence continue forever?
Work out the dimensions of the $n^{th}$ cuboid.
Can you explain how to make the $(n+1)^{th}$ cuboid from it?
What are the dimensions of the $(n+1)^{th}$ cuboid?
How could these ideas be used to calculate the sum $1^2+2^2+3^2+...+n^2$ quickly for any value of $n$?
Getting Started
Here is a picture of the third cuboid in the sequence. Can you see
how the sequence will continue to grow?
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Student Solutions
We received two very detailed solutions for this problem, one by Rajeev from Fair Field Junior School and one by Furkan from Amasya Science and Arts Centre.
Both correctly concluded that in order to build the 5 by 3 by 2 cuboid using the blue square prisms they would have to split one of the square prisms. Here is Furkan's pictorial representation of how to build the cuboid.
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Furkan continues:
We can also surround the above cuboid by six pink prisms. Again we have to split one prism as seen in the diagram below.
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Rajeev noticed this pattern continued for the next 4 cuboids in the sequence.
We discovered that:
$$ 6 \times 1^2 = 3 \times 2 \times 1$$
$$ 6 \times (1^2 + 2^2) = 5 \times 3 \times 2 $$
$$ 6 \times (1^2 + 2^2 + 3^2) = 7 \times 4 \times 3 $$
Rajeev then reasoned:
The method for forming the next cuboid is adding 6 square prisms and the dimensions of the prisms are the shortest dimension of the new cuboid. For example in the $7 \times 4\times 3$ cuboid the shortest dimension is 3 so the prism is 3 by 3 by 1. To form the next cuboid surrounded say by 6 yellow square prisms the dimensions of the prisms would be 4 by 4 by 1 and the dimension of the cuboid would be 9 by 5 by 4. The formula for the number of cubes to add on is $6 \times (n+1)^2$ i.e. 6 square prisms of side n+1.
The method for finding out the number of cubes used is $(2n+1) \times (n+1) \times n$ where n is the shortest side.
I also noticed that in each cuboid the width plus the height of the cuboid equals the length. So for example in the cuboid where 12 is the shortest side the cuboid has dimensions $ (2n+1) \times (n+1) \times n = 25 \times 13 \times 12 $.
The dimension for the $n^{th}$ cuboid is n, n+1 and 2n+1. And for the $(n+1)^{th}$ cuboid it is n+1, n+2 and 2n+3.
So we can write the $n^{th}$ cuboid as
$$6 \times (1^2 + 2^2 + ... + n^2) = (2n+1) \times (n+1) \times n$$
Thes ideas can be used to sum the series $ 1^2 + 2^2 + 3^2 + ... + n^2 $ quickly for any value of n using the formula:
$$ 1^2 + 2^2 + 3^2 + ... + n^2 = n(n+1)(2n+1)/6 $$
Teachers' Resources
Why do this problem?
This problem offers an opportunity to visualise in three dimensions and gives practice in working with sequences that do not grow linearly. It provides a possible introduction to the formula for the sum of the first $n$ square numbers. By considering how the cuboids grow from the $n^{th}$ to the $(n+1)^{th}$ the foundations are laid for learning about proof by induction.Possible approach
Set the scene for the problem - we are building up cuboids
from a sequence of square prisms, adding on six square prisms each
time.
Show an image of the first 3 by 2 by 1 cuboid and the second 5
by 3 by 2 cuboid. Give the students time to consider the first
problem - is it possible to make the second cuboid by adding the
six blue square prisms to it, without splitting any of them?
Different people visualise this in different ways so if isometric
paper and cubes are available, some students may wish to use them
to share their ideas.
Allow plenty of time for students to share in pairs and then
with the rest of the class their justifications for why at least
one of the square prisms needs to be split. Encourage those who can
do it by only splitting one
to share their methods.
Next, set the problem of continuing the sequence by adding on
the six pink square prisms. There is an image in the hint showing
the new cuboid formed. Again, the question is whether any square
prisms need to be split, and if so, how many. Explain that they are
trying to find a way to describe how to assemble the next cuboid
when six new pieces are added on, and give the students time to
investigate this. Encourage them to record their thinking in a way
that captures the way they "see" it.
Pause to reflect on what has been discovered so far:
$$6 \times 1^2 = 3 \times2 \times1$$
$$6\times(1^2+2^2) = 5\times3\times2$$
$$6\times(1^2+2^2+3^2) = 7\times4\times3$$
Ask them to predict how to continue this sequence, making
reference to the cuboids involved.
Once students have a consistent way of making the next cuboid
by adding on six square prisms, work can be done to express the
dimensions and the volume of the $n^{th}$ cuboid. Small groups
could produce a diagram and explanation to show how they would add
pieces on to make the $(n+1)^{th}$ cuboid.
Finally, once a formula for the volume of the $n^{th}$ cuboid
has been expressed, students can consider the relationship between
what they have found and the sum of the first $n$ square
numbers.
Key questions
Can you make the second cuboid by adding the six blue square
prisms? Will you need to split any?
Can you visualise a way of making the third cuboid from the
second, by adding the six pink prisms?
Can you visualise a way of making any cuboid from the previous
one in this way?
Are you sure you will always be able to make the next
cuboid?
Possible extension
Students could read
this article about proof by induction.
Sums of Powers shows another way of considering the sum of the
first $n$ square numbers.
Possible support
Lots of models of cuboids made from small cubes and lots of
diagrams annotated with dimensions can help students to build up a
picture of what is going on in this problem.
Try the problems
Picturing Triangle Numbers and
Picture Story for other tasks which think about sums of series
in a very visual way.