Tumbling probability
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| Cutting out your discs | |
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| You'll make three tumblers. | |
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| Align the tumblers on the cylinder. | |
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| Line up your lock! |
Warning: This activity requires cutting with sharp tools. Younger scientists should get an adult to help.
How easy would it be to guess a safe's combination? Build your own lock and find out!
You will need
Cardboard tube (9cm long)
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3 x sheets of thick card (at least 15cm x 15cm)
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1 x sheet of thin card
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Cardboard box
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Wooden or balsa wood dowel (15cm long, 10mm diameter)
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Plastic bottle top
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Craft glue
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Sticky tape
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Scissors or sharp blade
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Compass with pencil
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Black marker pen
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White paper
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Nail or drawing pin
What to do
Cut your cardboard tube into three 3cm sections.
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Use one to trace a circle in the middle of one of your sheets of thick card.
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Placing the point of your compass in the centre of this circle, draw a larger circle with a radius of 5cm around it. It should look a little like a CD.
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Cut out both circles, making sure to cut neatly around the outside of the line you drew for the inside circle. The cardboard tube will need to fit neatly through this hole. This makes your first disc.
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Repeat the process a second time (using the first disc as a template). For the third disc, do not cut out a middle hole – just cut the large circle. Its inner hole will need to be a lot smaller.
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From the scraps, cut two 2cm squares and a 2cm x 4cm rectangle.
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Make a 2cm cut in from the edge of each disc pointing towards the centre.
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Now for the gluing – take the two discs with the large holes in the middle. Slide one of the cardboard tubes through the hole of each disc until the disc sits around the middle of each tube. Glue them in place. You’ll still have a third section of tube, which you will need later.
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Take one of the two cardboard squares and slide it into the slot of one of the circles so most of it pokes out one side. Do the same with the other square, only placing it into the slot of the circle that has no hole in the middle. Lastly, place the cardboard rectangle in the slot of the last circle so half pokes out each side, gluing it in place. If you’ve done it right, looking at it edge-on, it will make a ‘+’ shape, while the other two will look like ‘T’ shapes.
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Cut a hole into the centre of your third disc that’s big enough for the dowel to pass through. Thread the disc 5cm down the length of the dowel and then glue it in place.
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These discs are your three ‘tumblers’.
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Roll the thin cardboard into a 9cm long cylinder wide enough to squeeze into the middle of your cardboard tube. If it’s too loose or too tight, the lock won’t work, so you’ll need to get the size perfect. When you’re satisfied that the tumblers can turn freely without spinning loosely, use sticky tape to fix it in place.
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The lock will need a case to sit in. Cut the two opposite corners from a cardboard box, giving you two smaller quarter-boxes that are 15cm x 10cm x 10cm. Cut a hole in the wall of one box that the dowel will fit through.
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Glue one end of the thin cardboard cylinder to the inside of this hole. The dowel will need to pass through the cylinder and out through the hole later.
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Remember the left-over piece of cardboard tubing? Thread it onto the cylinder to act as a spacer. Thread the ‘T’ tumbler with the large hole next, with the square pointing away from the spacer. Then thread the ‘+’ tumbler, followed by the second ‘T’ tumbler with the square pointing towards the ‘+’ tumbler. Lastly, insert the dowel down the length of the cylinder and through the hole in the wall. Your tumblers should line up so the cards are able to push against each other if they are turned.
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Take the second corner of the box and slide it into the first so they overlap. Make a hole in its wall for the other end of the dowel to poke through. Glue the dowel in place.
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Draw or print out a clock-face on your sheet of white paper. Cut it out, and cut a hole in the centre for the dowel. Glue it on the outside of the box around the dowel so you have dial for your lock.
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Nail or tack a plastic lid to the end of the dowel.
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Turn the dial twice clockwise to line up your tumblers. Use your black marker to draw a line on the plastic lid pointing up at the twelve.
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Turn your dial clockwise so the black line points to the 3, then anticlockwise one complete turn and then onto the 11, and then clockwise again to the 6. Draw a line on each tumbler to mark this ‘unlocked’ position.
What’s happening?
Combination locks like these ones require a code with a string of digits that look something like ’3 – 16 – 7’. To unlock, the dial needs to be turned clockwise all the way around at least twice until the dial is pointing straight up (at the ‘twelve o’clock’ position). This ensures the tumblers are lined up correctly.
The first turn takes the dial from the starting position to point at number 2. It pushes the back tumbler so its gap (our black line) lines up with a release mechanism, or the latch. The second turn goes back the other way – anticlockwise – five places. This pushes the middle tumbler so its gap lines up as well. The third turn goes clockwise again by four places, rotating the front tumbler so all three gaps are together. This forms a space for the release mechanism to lift into, allowing you to lift the latch and open the lock.
Old fashioned safe crackers could open locks by listening carefully to the tumblers rolling against the release mechanism. Of course, combination locks on high-security safes are far more complicated today, often relying on electronic locks rather than mechanical ones.
Applications
How lucky would you have to be to guess the right combination on a safe? It depends on the numbers.
In our safe, there are twelve numbers, like the positions on a clock. If twelve people each guessed a different number, then one of them would be right, but the rest would be wrong. There is a one-in-twelve chance of getting it right by luck alone. What about guessing two numbers?
If each person had their own cage containing twelve magical number-guessing mice, there’d be 144 mice altogether. In each cage, one mouse would correctly guess the second number. But only one person guessed the first number as well. That means there is one mouse out of 144 who guessed the right number AND belongs to the person who also guessed the first number.
What about three tumblers? Imagine each magical mouse has twelve magical fleas…
Do you see the pattern? Three tumblers with twelve numbers means 12 x 12 x 12. We can also write this as 123, or simply 1728. You’d need to try that many different combinations to make sure that you could get into the safe.
