# Start Cube Drilling

Imagine a 3 by 3 by 3 cube. If you and a friend drill holes in some of the small cubes in the ways described, how many will have holes drilled through them?

This activity is all about
imagining, like you might when you listen to a story or poem.

There is no need to write or draw
anything

BUT talking about it could be
good!

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Imagine a 3 by 3 by 3 cube hanging in front of you with just the front face facing you.

The cube is made up of 3 x 3 x 3, that is 27, small cubes.

You drill a hole through the four corner cubes, that are facing you, all the way through to the back.

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A friend looks down on the cube, from above, and they also drill four holes through their four corner cubes all the way through to the bottom.

You and your friend examine all the 27 small cubes.

QUESTION

How many small cubes will then
have holes drilled in them?

If you'd like more of a challenge, try Cube Drilling .

When you drill your holes, can you picture which small cubes have
got holes in them?

How many do you drill holes in?

Now think about your friend.

Can you picture the small cubes that he or she drills holes in?

Are any of the cubes your friend drills ones that you've already drilled?

Are some of these cubes ones that you haven't already drilled?

How many do you drill holes in?

Now think about your friend.

Can you picture the small cubes that he or she drills holes in?

Are any of the cubes your friend drills ones that you've already drilled?

Are some of these cubes ones that you haven't already drilled?

We had a few solutions come in from pupils around the world showing how they came to their answers. Firstly, Lesley from Kings Road Primary School in England, wrote;

16. When I drill my holes, I drill through three cubes at a time through each corner. When my friend drills through the top, he only drills one new cube at each corner because I have already drilled the rest.

Here is a solution by Louis, Katherine, Angus, William, Isobel and Harrison from St. Aiden's School England:

We started by making a cube from 27 Unifix cubes and we checked that it really was a 3 x 3 x 3 cube. We then imagined that we drilled a hole through the cube from the front face into each of the four corner cubes, all the way to the back, marking the cubes with the holes in with bits of blutak. 12 of the cubes now had a hole in them. That is because on the front face we made 4 holes and each one went through 3 cubes until it got to the back. Then we looked down from the top face and repeated this and we marked all the 12 cubes with blutak. So we had made 24 holes in the cubes. But we found that some of the cubes had been marked twice and these were the corner cubes. Because a cube has 8 corner cubes, that meant we had to subtract 8 so we now had 24 - 8 = 16 cubes with holes in them.

We then thought about what would happen if we had a cube that was 4 x 4 x 4. This would have 64 small cubes. Again we started by drilling from the front and made 4 holes and this time they went through 4 cubes each so there were 4 x 4 = 16 cubes with holes in. If we did the same from the top there would be another 16 cubes with holes in so that is 32 altogether. But the corner cubes have 2 holes so we need to subtract 8 again so the total number of cubes is 32 - 8 = 24.

We think that whatever number you choose for making your cube (eg 3 x 3 x 3, or 4 x 4 x 4), you can do the same thing. You multiply the number by 4, then double it (front face and top face) and then subtract 8 (for the corners that have 2 holes) and that is the total number of holes you have drilled.

Thank you for that thorough explanation

Then Grace from Ohaupo Primary School in New Zealand sent in this:

I made a cube out of blocks and tallied up the bottom rows first, I got an answer of 6. Then there were the top rows, I also got an answer of 6 then. Then I had to drill in the friend's view. I added those up and got an answer of four. I then added them to my answer of twelve. The answer I got overall was 16. So 11 cubes were not drilled.

Rik from Gdgoenka World School in India sent in the following:

The answer is 16, as we need to drill to the back first we drill 12 cubes, 3 for each back multiplied by 4 corners and we get 12. Then when your friend drills from up only 4 cubes are drilled. As 12 + 4 = 16 the answer is 16.

Finally Amran From Prioslee Primary Academy England:

So I imagined a 3 by 3 by 3 cube and thought if you drilled four holes and then another four holes you would add 4 + 4 which then equals 8 meaning there were 8 holes drilled in from the front and altogether looking from the front and the back there would be 16 holes.

Thank you all for these submissions. If you are looking at these with the idea of exploring the challenge further you should find their comments useful.

### Why do this problem?

The whole idea of this problem is to invite children to picture something in their mind and in this instance, pupils will need to be familiar with properties of a cube. Ideally, it would be good to encourage your class to tackle this challenge purely by trying to imagine what is happening. To convince you and each other of their solutions, they will need to explain particularly carefully what they are picturing, which can be quite tricky, and you may find that they gesticulate rather a lot! In order to reach a joint conclusion, you might find it helpful to make a model of the cube from interlocking cubes.### Possible approach

To introduce the activity, it might be worth simplifying the problem by starting with a smaller cube (say 2 by 2 by 2). Invite them to picture this cube in their heads and ask questions, such as:- How many small cubes are there altogether? [8]
- How many corner cubes? [8]
- What other kinds of small cubes make up the cube? [none - they are all corner cubes]

You could then move on to picturing a 3 by 3 by 3 cube and probe in a similar way before trying this problem.

Introduce the challenge itself orally, giving each piece of information one at a time and pausing between each. At this stage, ask the class to work independently of each other. Once you have posed the question, explain that you'd like learners to continue to work alone for a short time. Give them chance to work on the solution.

After a suitable period of time, invite them to share their ideas with a partner. Warn them that they will have, say 15 minutes in order to reach a consensus and to be able to explain their reasoning to others. You could film pairs' explanations and watch a few as a whole class in the plenary, or during a subsequent lesson.

Introduce the challenge itself orally, giving each piece of information one at a time and pausing between each. At this stage, ask the class to work independently of each other. Once you have posed the question, explain that you'd like learners to continue to work alone for a short time. Give them chance to work on the solution.

After a suitable period of time, invite them to share their ideas with a partner. Warn them that they will have, say 15 minutes in order to reach a consensus and to be able to explain their reasoning to others. You could film pairs' explanations and watch a few as a whole class in the plenary, or during a subsequent lesson.

### Key questions

Try to tell me what you are picturing in your mind.What happens during the first drilling?

Can you picture which cubes have holes now?

What happens during the second drilling?

Can you picture which cubes have holes now?

### Possible extension

If your class would like more of a challenge having tried this problem, they could look at Cube Drilling.