Picture a pyramid
Imagine a pyramid which is built in square layers of small cubes. If we number the cubes from the top, starting with 1, can you picture which cubes are directly below this first cube?
Problem
This activity is about imagining a picture in your head. You will probably have to write some numbers down, but try not to draw anything.
Imagine a step pyramid. Each layer is made from a number of stone cubes.
There is a square number of cubes in each layer, which increases for each layer from top to bottom.
The cubes are numbered starting from the top layer so that, looking down from above, each new layer carries on counting starting from the North West corner. (When the end of a row is reached you start at the West end of the next row.)
There are equal steps on all four sides of the pyramid (in other words, each successive layer "sits" exactly in the middle of the layer beneath it).
We'll imagine a six-layered pyramid that uses 91 cubes in total.
What numbered cubes are vertically underneath (in a straight line):
Now imagine stepping up two layers at a time, starting at the middle cube on the South side, second layer up. Which numbered cubes would you step on to end at the top, if you stepped up in a straight line?
Starting at the South West Corner and stepping up one layer at a time on the corner steps, which numbered cubes would you step on?
What if a second pyramid was built, numbering the cubes as they built the pyramid, so the numbering would start from the bottom North West corner and starting the second row as before.
So number 1 at the bottom and the last cube number 91.
What numbered cubes are vertically underneath (in a straight line):
Is there a cube that has the same number in both pyramids?
What was the difference for you in doing the second pyramid compared with doing the first pyramid?
Getting Started
Can you picture how many cubes are in each layer of the six-layered pyramid with 91 cubes in total?
Which layers will have a cube which is directly below cube 1?
Which layers will have a cube which is directly below cube 4? And cube 8?
You might find it useful to record the layers somehow.
Try comparing the positions of the cubes in the first and second pyramids. What do you notice?
Student Solutions
This is a tricky challenge but we had some good responses. Not all of them were completely correct, but most contained excellent thinking and so here are a selection. If you want to know the 'answers', then scroll down to the bottom of this page.
We had these solutions from C of E Primary School, Ledston. Firstly, Mia and Millie said:
We got 10 and 43 as the cubes directly underneath cube number one by:
• Counting all the cubes on each layer. Which led us to a pattern of 1,4,9,16,25 etc.
• Imagining it clearly in our heads
• Counting each cube in our heads as we went down
• We wrote the number of each layer like this and put a tick next to the ones we knew it would be (the third and fifth layers)
For the cubes directly beneath 4 we got 24 and 67, and directly beneath 8 we got 39 using the same strategies as before.
Then from Edward, Linden and Sam at the same school we had:
To get the first answer we did:
First we worked out how many cubes are on each layer
Layer 1=1 layer 2=4 layer 3=9 layer4=16 layer 5=25 layer 6= 36
Visualise how many cubes there are so far going from the top to bottom
Layer 1 =1 layer 2=5 layer 3=14 layer 4=30 layer 5= 55 layer 6=91
Then we found out which layers have a centre.
We found the layers 3 and 6 have centres.
We then find out what cube was in the centre
The answers we got were 10 and 43.
To get the answer for the 2nd question we did:
First we worked out how many cubes are on each layer
Layer 1=1 layer 2=4 layer 3=9 layer4=16 layer 5=25 layer 6= 36
Visualise how many cubes there are so far going from the top to bottom
Layer 1 =1 layer 2=5 layer 3=14 layer 4=30 layer 5= 55 layer 6=91
Then we found out that it had to be on a layer with an even number
These were the answers we got 24 and 75
To get the third answer we did:
First we worked out how many cubes are on each layer
Layer 1=1 layer 2=4 layer 3=9 layer4=16 layer 5=25 layer 6= 36
Visualise how many cubes there are so far going from the top to bottom
Layer 1 =1 layer 2=5 layer 3=14 layer 4=30 layer 5= 55 layer 6=91
Find a layer with a number that is in the 8 times tables.
We got the layer 4
We got the answer 33
Finally from Adam and Patrick
This is how we completed the nrich pyramid problem step by step.
First we read the instructions until we understood the problem.
Next we counted the number of squares in each layer starting from the North West corner going right and added them together as we went along; the answers were: 1+4=5+9=14+16=30+25=55+36=91.
We then looked at the first question which was: what number cubes are vertically underneath (in a straight line):
Then we attempted to solve the first question.
Soon we realized that the numbers went in a straight line through the middle of the pyramid.
We found out that only layers that worked were odd.
Those were: layer 3 & 5 and the numbers: 10 & 43
After we solved that question we moved on to the next which was the same question just a different starting number.
The next question was harder but we solved it as well.
The answer was: 53 and 13.
The next few answers were similar.
At that point we were on a roll.
For the last 4 questions we worked out they were the same as the first three but the last one was simple.
The answer was no.
Here are the solutions -
Under 1 is 10 (followed by 43), under 4 is 24 (followed by 76), under 8 is 39.
Middle of south 2 up is 53 and 2 at a time takes you to 13 and 1.
South west corner stones are 86, 51, 27, 12, 4 and 1.
Re-numbering the other way gives 82 under 91, under 89 it's 71 and under 80 it's 45.
There are no numbers in the same place with the re-numbering.
But again well done for trying this unusual harder challenge that is purely based on visualising.
If anyone has a further try do send in your solutions.
Teachers' Resources
Picture a Pyramid ...
Imagine a step pyramid.
Each layer is made from a number of stone cubes.
There is a square number of cubes in each layer, which increases for each layer (increasing from top to bottom).
The cubes are numbered starting from the top layer so that, looking from above, each new layer carries on counting starting from the North West corner. (When the end of a row is reached you start at the West end of the next row.)
There are equal steps on all four sides of the pyramid (in other words, each successive layer "sits" exactly in the middle of the layer beneath it).
We'll imagine a six-layered pyramid that uses 91 cubes in total.
What numbered cubes are vertically underneath (in a straight line):
Now imagine stepping up two layers at a time, starting at the middle cube on the South side, second layer up. Which numbered cubes would you step on to end at the top, if you stepped up in a straight line?
Starting at the South West Corner and stepping up one layer at a time on the corner steps, which numbered cubes would you step on?
What if a second pyramid was built, numbering the cubes as they built the pyramid, so the numbering would start from the bottom North West corner and starting the second row as before.
So number 1 at the bottom and the last cube number 91.
What numbered cubes are vertically underneath (in a straight line):
Is there a cube that has the same number in both pyramids?
What was the difference for you in doing the second pyramid compared with doing the first pyramid?
Why do this problem?
This problem requires pupils to visualise the described pyramid in order to answer the questions. Your class will also be using knowledge of square numbers to solve the problem.
Possible approach
It might be best to introduce this activity orally so that children do not have to read lots of text. You could jot down a few reminders on the board for them to refer to as they try each part of the problem. Before asking them to picture the cubes vertically above and below each other, you might like to ask pupils to tell you where particular numbered cubes are in the pyramid so that you can check everyone is imagining the same thing. You might suggest that at first they try to answer your questions without writing anything down, and talk about any difficulties this poses, but soon give them the opportunity to record the pyramid in whatever way they find most helpful. Share some of these representations with the whole group before going on to try the problem itself.
This sharing is a very valuable part of the visualisation process as it is a skill in itself not only to describe your own visualisation, but also to understand someone else's and compare it with you own.
Key questions
Possible extension