Holes
I've made some cubes and some cubes with holes in. This challenge invites you to explore the difference in the number of small cubes I've used. Can you see any patterns?
Age
5 to 11
Challenge level
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Here we have three solid cubes and three cubes that have holes.
They're just the first three in a series that could go on and on.I was wondering about the number of cubes used in each ...
Then I thought about the difference between those numbers.
So, for example, I found that the first cube, $3$ by $3$ by $3$, used $27$ cubes.
The same cube with holes used $20$ cubes, a difference of $7$.
You can call these types of cubes what you like but I called them "Solid" and "Frame".
How about exploring the numbers for the next few Solid and Frame cubes?
Do you notice any patterns?
Can you explain any of the patterns?
How will you record what you find?
You might find interlocking cubes useful, or perhaps computer drawing programs, or even spreadsheet software?
You might find interlocking cubes useful, or perhaps computer drawing programs, or even spreadsheet software?
We had a few solutions sent in for this challenge. I think that maybe there were other pupils who got so far, on their own or with the teacher, and decided it was not worth sending in. We would like to reassure you that all journeys in the solving of challenges - however far you get - we welcome!
Here is the excellent result that we received from class $5$A at St. Christopher's School in Penang, Malaysia.
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Well Done !
From Jenthghe and Olivia we received.
For this problem we used a trial and error method!
For solid cube number $1$ we did, $3$x$3$x$3$=$27$
For frame cube number $1$ we did $3$x$3$x$3$=$27-7=20$ we took away $7$ because there was seven holes in the cube.
We did the same for solid and frame cube number $2$ but this time we found that we had to do $4$x$4$x$4$=$64$ (for the solid cube), then find what the frame cube had which was $4$x$4$x$4$=$64-24=40$ (for the frame) we took $24$ away because that was how many holes were in the cube.
For solid and frame number $3$ we did $5$x$5$x$5$=$125$ (for the solid cube) and for the frame we did $5$x$5$x$5$=$125-65=60$. we took away $65$ because there were $65$ cubes missing.
This was a good start to this challenge but a closer look at the frame cubes would be necessary. Then you might see that the $24$ you take away from $64$ cubes for the second cube only takes account of the $4$ cubes on each face that were removed. There was also some cubes removed from the centre of the solid cube and they would form a $2$ by $2$ by $2$ cube. This would mean a total of $32$ cubes altogether.
Shriya from the International School Frankfurt in Germany sent in this:I used Lego blocks to build the cube frame.For the 4×4×4 cube, I got 32 cubes with holes and 64 without. For the 5×5×5 cube, I got 44 cubes with holes and 125 without. For the 6×6×6 cube, I got 56 cubes with holes and 216 without. After this, I found out the pattern. There was a difference of 12.
12 × (n - 3) + 20. I confirmed this by building a 7×7×7 cube and my guess was correct. I got 68 cubes with holes and 343 without. Now, with the formula above, I can predict how many cubes I will need to make a cube frame.
Thank you for these solutions.
Holes
Image
Here we have three solid cubes and three cubes that have holes.
They're just the first three in a series that could go on and on.I was wondering about the number of cubes used in each ...
Then I thought about the difference between those numbers.
So, for example, I found that the first cube, $3$ by $3$ by $3$, used $27$ cubes.
The same cube with holes used $20$ cubes, a difference of $7$.
You can call these types of cubes what you like but I called them "Solid" and "Frame".
How about exploring the numbers for the next few Solid and Frame cubes?
Do you notice any patterns?
Can you explain any of the patterns?
Why do this problem?
This problem gives opportunities for pupils to explore, to discover, to analyse and communicate. It's a real catalyst for pupils' curiosity. It allows pupils to approach it in whatever way they find most helpful. It also provides opportunities for using and extending visualising skills. The activity also opens
out the possibility of pupils asking “I wonder what would happen if . . .?” showing their resilience and perseverance.
Possible approach
You could show the group models of the first solid cube and first frame cube. Invite them to talk to a partner about them and to share their observations with the whole class. You could then ask pairs to suggest what the 'next' two cubes would look like. You could also ask them to explain why they think you started with a $3$ by $3$ by $3$ cube rather than, say, a $2$ by $2$ by $2$
one.
You may like to challenge learners to find the number of small cubes each is made up of and they could share their methods for doing so. Some may need to make their own model to help them, but others will be able to calculate the number of cubes using certain facts they know. By sharing different methods, some children may take on a new method because they find it works better for them
compared with the original way they chose. Pupils can then work on the challenge itself in pairs or small groups.
Give the class the freedom to choose their own way of recording their findings and share these in the plenary, as well as sharing results. Encourage explanations of the number patterns, rather than just 'pattern spotting'.
Key questions
Tell me about about what you've found.
Are there other ways of showing what you've found?
Possible extension
Learners could explore what happens when they count the square surfaces that are visible on each small cube.
Possible support
Having plenty of interlocking cubes available will aid some children, whereas others may wish to use computer drawing programs and/or spreadsheet software.