Maths Week Scotland 2021 - Problem 4 - Rods

Chris Smith - This problem is all about measuring different lengths.

Now normally we would measure with a ruler or a metre stick but we're not going to do that today.

Instead I've got three rods of different lengths. 9, 3 and 1 unit long.

And we can use these in different combinations to measure different lengths.

For example, if I want to measure 10 units I can take the 9 units and put the 1 rod at the end. We know it makes a rod 9 + 1 = 10 units long.

If we want to measure 7 units, I can take that 10 then put the 3 unit rod next to it. 9 + 1 = 10. And 10 - 3 = 7.

Can you work out how to arrange the three rods to measure all the unit lengths from 1 to 13?

And for bonus points, you can add a fourth rod that will allow you to measure every unit length from 1 all the way up to 40 units. What length would that fourth rod need to be?

First of all, it’s important to remember that you don’t have to use all the rods for every length. You could use one rod, or you could use two, or you could use all three.

And remember you can use these in different combinations to allow you to add to make longer lengths or to subtract when we work out the difference.

You might even want to make your own rods, maybe using building blocks or cutting them out from card to help you to crack this problem.

Give it a go.

Chris Smith: Did you figure out how to arrange the rods to measure those different lengths?

It basically becomes a question of adding and subtracting.

Let's take a look at how I solved this one.

Length 1 is dead easy. There we go. We know that is 1 unit long.

If we bring in the 3 rod we can see the bit that's remaining is 2 units long.

3 - there we go. And to make 4 we just add in the 1.

To get 5 we'll put the 9 next to this 4. That means this bit here must be 5.

To make this section longer again we take away that 1 and so this section here is 9 take away 3 gives us 6.

To make 7, first of all make 10, take away 3. This section here is 7.

To get 8 what we do is take the 9 and we subtract the 1 and so what's left here must be 8.

To make 9 we just use the 9 rod.

10, we add in the 1, so that is now 10.

To make 11 we start with 3 and 9, which gives 12, and then we subtract 1, so this section here must be 11.

Take that 1 away, this is 12.

And how do we get 12? We combine all three rods and that gives us 13.

Did you work out how long the fourth rod would be to allow us to measure every unit length from 1 all the way up to 40?

We know the other three rods add up to 13. So there's just one possible answer if we can only add one rod that can take us up to a length of 40 units, You have to be able to add that rod onto the 13 to make 40.

So the rod must be 40 take away 13, which equals 27 units long.

And now we use a similar process to before. We already have all of the lengths from 1 to 13. And we can use these rod combinations with the 27. Either adding to 27 or subtracting from 27 to allow us to get every number from 1 all the way up to 40.

We already know how to form the number 13. We put the 9 and the 3 and the 1 together, and that's 13.

So if we line this up with the 27, if this bit's 13, this bit must be 14 because 27 - 13 = 14.

If we want to make 15 we just remove the 1, and then we've got 27 - 12 = 15.

And we can keep going through all of the numbers from 15, 16, 17 all the way up to 40, which is 1 plus 3 plus 9 plus 27.

Did you spot anything interesting about the lengths of the rods in this question? They were all powers of 3: 1, 3, 9 and 27. And that's no coincidence because you had three options for each rod. You could add them, you could subtract them or you could leave them out altogether.

So if you're dead keen you might want to go and find out more about base three numbers.