How Safe Are You?

Age

7 to 11

Challenge level

How much do you have to turn these dials by in order to unlock the safes?

Problem

We're going to look at opening safes!

Many have dials on them, and you turn the dials to open them, like in the picture below.

A metal safe with a hand turning the dial.

Here's a simple dial with just the numbers $0 - 5$ on it.

A dial from a safe with the numbers 0 to 5 clockwise around the outside, and an arrow at the top pointing to 0.

To open the safe, the dial has to be turned so that the number $2$ is next to the arrow, like this:

A dial from a safe with the numbers 0 to 5 clockwise around the outside, with 2 at the top.

How much was the dial turned to get the $2$ at the top?

The safe below has a different dial with the numbers $0 - 7$ instead.

How much does the dial have to be turned to get between these two pictures, so that the number $5$ is at the top?

A dial with the numbers 0 to 7 clockwise around the outside, with 0 at the top.

A dial with the numbers 0 to 7 clockwise around the outside, with 5 at the top.

The number that you have to get at the top is often called the "combination" of the safe. These three different safes all start with $0$ at the top. You have to find the amount of turning to get to the combination, shown in each second picture:

Two dials with numbers 0 to 11 around the outside going clockwise. The first dial has 0 at the top and the second dial has 3 at the top.

Two dials with numbers from 0 to 8 around the outside. The first dial has 0 at the top and the second dial has 3 at the top.

Two dials with numbers from 0 to 23 around the outside. The first dial has 0 at the top and the second dial has 6 at the top.

Now have a go at these different safes. Remember they would all start with $0$ at the top. How much does each one need to be turned to get to the combinations shown below?

Image text description

5 dials labelled A to E. Dial A has numbers 0 to 23 with 9 at the top. Dial B has numbers 0 to 7 with 6 at the top. Dial C has numbers 0 to 5 with 4 at the top. Dial D has numbers 0 to 11 with 5 at the top. Dial E has numbers 0 to 8 with 5 at the top.

Student Solutions

George and Dominic from St Nicolas C of E Junior School sent us very clearly explained solutions to this problem. Thank you! Here is what they wrote for the first part:

1. For the first question I worked it out by seeing that number 3 was at $180^\circ$ and that meant I needed to divide 180 by 3, giving the angle in between each number. The answer was $60^\circ$. By knowing this I could work out that number 2 was $120^\circ$ away from zero anticlockwise and was $240^\circ$ clockwise from zero.

2. To work this out I did almost the same as I did with question 1 except that I divided 180 by 4 instead. This gave me $45^\circ$ between each number. By multiplying $45^\circ$ by 3 (which is how many numbers 5 is away from zero ) I got $135^\circ$. This meant that I would have to either turn the dial $135^\circ$ clockwise or $225^\circ$ anticlockwise.

3. This was a little easier to work out because number 3 was a quarter of the way round the dial meaning to get to it I either had to turn it $90^\circ$ anticlockwise or $270^\circ$ clockwise.

4. This was harder than the previous questions because there wasn't a number at the $180^\circ$ point. To work it out I divided $360^\circ$ by 9. This gave me how many degrees I would have to turn the dial to get to the next number. The answer was $40^\circ$. This showed me that to get to number 3 I had to turn it either $120^\circ$ anticlockwise or $240^\circ$ clockwise.

5. At first sight this looked really tricky but after thinking about it wasn't. Number 12 is $180^\circ$ away from zero. Half of 12 is 6 and half of 180 is 90, meaning that to get to the number 6 I have to either turn the dial $90^\circ$ anticlockwise or $270^\circ$ clockwise.

George and Dominic continued to explain how they had gone about the second part of the solution:

The next five were the hardest questions because we couldn't see where the zero was at the start.

A. For this question I worked out that 12 was $180^\circ$ from zero and 180 divided by twelve is 15. This means that each section is $15^\circ$ wide. There are 9 spaces between zero and nine which means that to work out the answer we have to times 9 by 15 which equals 135. So to get to 9 I have to turn the dial either $135^\circ$anticlockwise or $225^\circ$ clockwise.

B. We found this one quite easy because again 6 was a quarter of the way round, so this meant that we had to turn the dial $90^\circ$ clockwise or $270^\circ$ anticlockwise.

C. First I worked out that 360 divided by 6 is 60 meaning that each section is $60^\circ$ around the dial. Using this information I worked out that I would have to turn the dial either $120^\circ$ clockwise or $240^\circ$ anticlockwise.

D. On this one I knew that 6 was at $180^\circ$ but you had to work out what 5 was. So I did 180 divided by 6 which equals 30. Then I did 180-30 which equals $150^\circ$ so it was $150^\circ$ anticlockwise and $210^\circ$ clockwise.

E. Firstly for this I divided 360 by 9 which gave me 40. I then multiplied 40 by 5 which gave me 200. Knowing this I could work out that to get to 5 I either have to turn the dial $200^\circ$ anticlockwise or $160^\circ$ clockwise.

We ended the lesson by writing down 3 instructions for someone else to follow which would give our combination. Sophie, another girl in our group, wrote down: Using dial D, turn the dial $240^\circ$ anticlockwise. Then from that point, turn the dial $210^\circ$ clockwise. Then from this point, turn the dial $90^\circ$ anticlockwise. What's my combination?

We worked out that it was 8 1 4.

Very well done to you both. You obviously worked hard on this activity. Thank you too, to Eve and Rachel from Castle Carrock Primary who also sent in well-explained solutions.

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How Safe Are You?

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