As promised, I will give you two different answers. The first method has a slight drawback in that each lock can be used only once. The second method avoids this problem by not sending any keys through the mail.

Alice places the message into the box and locks it with a padlock. She then sends the locked box to Bob. Neither Bob nor Charlie can open the box, but Bob can take possession of the Box and put it in a safe place.

Bob then sends a plain letter back to Alice informing her that he received the box and it is securely stored away. Once Alice receives the letter she sends the key to Bob. Because the key is of no use without the box, Charlie can’t use the key to get to the original message. When Bob receives the key, he can take the box out from the safe place, open it, and read the secret message.

This method has a drawback. After the message is exchanged, Charlie had access to the key. Assuming that Charlie will make an effort when it comes to sniffing around, he might make a copy of that key. The next time Alice and Bob want to exchange a message, they will need to use a different padlock. So each lock and key can only be used once.

This method avoids the drawback of the first method.

Again, Alice starts by placing the message into the box and locking it with a padlock. Just like before, she sends the locked box to Bob. Neither Bob nor Charlie can open the box. Now instead of putting the box away, Bob adds another padlock to the box and sends the

Now instead of putting the box away, Bob adds another padlock to the box and sends the box with both locks secured back to Alice. When Alice receives the box back, she can now remove her lock and send the box to Bob again. This still leaves the box secured with Bob’s lock, and Charlie will not be able to open it. Bob now receives the box with only his lock, so he can open the box and read the message.

Charlie will never be able to open the box because it is locked with at least one lock whenever it is in transit. On top of that, the keys to the locks are never exchanged. This means that Charlie will never have access to the keys a thus will never be able to copy them. So the next time Alice and Bob want to exchange a secret message they can use the same locks.

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Fortunately, Alice has a box into which she can place the message. The box can be secured with one or more padlocks. Unfortunately, they can’t send the key together with the box because Eve would then be able to open the box with the key and read the message.

Which method should Alice and Bob use to send the message?

This riddle is a relatively easy logic question, and I can think of two simple solutions to this problem. Can you find more?

I will publish and discuss the answer tomorrow.

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The obvious answer is of course that **the bear is white**. The explorer started from the north pole. He first walks a mile south, so now he is one mile away from the north pole. When starts walking west, he walks along a circle around the north pole. At each point along the circle, he is exactly one mile away from the north pole. When he finally walks north again, he walks along the radius of the circle back to its centre at the north pole. This means that the wild bear must be a polar bear.

If we think of other places that the explorer might have started, we find that he could have also started near the south pole. Of course, there are no polar bears at the south pole, only penguins.

To find out where exactly, we start by looking at the westward leg of the journey. If the explorer is near the south pole, then the walk due west prescribes a circle around the south pole. One of those circles will have a circumference of exactly one mile. So the mile long walk due west will return the explorer to the start of that leg. If we go north one mile from there, we find the starting point of the explorer. Now, because the westward leg could have started anywhere on the circle the possible starting points also prescribe a circle around the south pole. I will let you figure out the radius of that circle for yourselves.

We already have infinitely many starting points along a circle around the south pole. But we can find even more! Instead of walking around the south pole only once, the explorer could have walked around the south pole 2 or 3 times or more. If he walks twice around the pole, we find a circle with a circumference of half a mile. If he walks three times around the pole, the circle must have a circumference of one-third of a mile, and so on. For each of these circles, the possible starting points of the explorer is a circle with a radius that is one mile larger.

Can you calculate the radii of these circles? Post your answers in the comments.

]]>In our *Weekly Puzzle* section, we will be teasing your brains with a different mathematical or logical puzzle every week. Let’s start with a classic from the great Martin Gardner. According to Gardner, the question is based on an old riddle. He points out that the obvious solution might not be the only one.

So here comes the riddle. Be sure to check the answer which we will be publishing tomorrow.

An explorer is out exploring. He decides to go for a walk to look at his surroundings. He walks one mile due south. He then turns right and walks one mile due west. After that, he turns right again and walks one mile due north. The explorer finds himself exactly at the point from which he started. Suddenly he spots a wild bear. What colour is the bear?

Can you find the answer?

Can you think of other places that the explorer could have started? Hint: there are no wild bears in those other places.

In the original puzzle, the explorer shoots the bear. Nowadays we don’t want to kill the wildlife unnecessarily. So we decide to simply watch the bear, take a few nice photos of it and otherwise leave the animal unharmed.

]]>The app has been completely redesigned and re-structured. Everything has been optimised to make it easier for you to find the information you need. This will hopefully make your revision more successful.

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Mikail