The Magic of Logic (14)

On elevators and dishonest couriers

With the pandemic flaring up again, in Europe, interaction with the beloved grandkids might soon be restricted. Once again. Of course we are fully vaccinated, but they are too young, and vulnerable. I predict that schools will be soon shut down. Once again.

Thank heavens for adventure playgrounds, which they love to visit — like recently after school. On this occasion, I had a new puzzle for the two:

A man lives in the 12th floor of an apartment building. Most evenings, when he comes home, he takes the elevator to the 10th floor, and then walks up the last two flights. Why?

Enders and Hennes had some immediate ideas: perhaps the elevator only goes up ten storeys? No, I said, sometimes he goes straight to the twelfth. He does it for exercise? No, even when he is dead tired, he’ll often take the elevator to the tenth and climb the stairs.

Well, the playground has so many attractions, and they went about their rambunctious activities, running, jumping, climbing. But occasionally Enders, 9, would come to me with a new idea: “Does he have an aunt living on the tenth floor, and stops to feed her cat?” No, Enders. It is a straightforward logical problem, and when you find the solution you will know you have got it. Nothing roundabout and complicated…

The interesting part is that he kept interrupting his play, every ten or fifteen minutes, to present a new theory. It means that the problem was in his mind all the time. He was thinking. And becoming a little frustrated. On the drive home he tried one or two more ideas, and I said to him: “Listen, if you don’t solve it, you get dry bread and vinegar for dinner.”

That put him into deep thought mode. And just before we arrived, he suddenly yelled out: “Got it! Great problem.” And he gave me the solution I was looking for.

In the weeks after this occasion I have given the problem to maybe twenty people — to some of the best chess players on the planet, to young super-talents, to problem specialists, scientists, normal people. And it stunned me to see how most struggled with the solution. My inbox is full of abstruse, far-fetched explanations. I considered this a fairly simple problem. How could so many people have such difficulty solving it?

So I started giving them a hint: Usually this man goes to the tenth and then walks up the last two floors. But if he comes home with friends, they always go straight up to twelve. That clicked for some — but many still struggled. I will give you the solution at the end of this article.

Enders and Hennes got a fine dinner that evening, and were up for another logical problem. So they got the following.

The dishonest courier

We need to send a valuable item to an associate in a different city. There is a very fast, very inexpensive courier service. Unfortunately, the couriers themselves are extremely dishonest and will steal anything that is in the box we use to send the goods.

Luckily the box can be secured with padlocks (as shown in the picture), making it impossible for the courier to get at the contents. But of course, our partner in the other city does not have the keys to our padlocks.

How can we send the goods safely?

Here are some points to note:

• We can only use regular padlocks, of which everone has an ample supply.
• We do not have a combination lock, where we could transmit the numbers by phone.
• We do not use any other form of delivery, e.g. send the key by regular post.
• The courier service is very inexpensive, we can use it as often as we like.
• Sending a key in an unlocked box or attaching it to the outside of the box is pointless: the courier steals keys.
• Transmitting information on the shape of the key, e.g. by email or fax, doesn’t work either: our associate does not have the tools or the skill to quickly manufacture a key.

The solution is not a silly little trick, but straight-forward and very satisfying. If you do not think to yourself “Wow, that is so simple and so elegant!” you have not yet found the correct strategy.

This problem was too hard for Hennes, 8, but Enders had some immediate ideas that did not work. After three minutes his face lit up: he had the solution, and knew he had to be right. It is simple and elegant.

I am not going to tell you the answer to this one. But before I wrap up with the elevator solution, there is one more thing to relate.

Since a few weeks we have been a visitor in the garden, a brown squirrel we call Billy. Together with the kids I have been putting out nuts, which Billy faithfully collects, two or three time a day.

We keep placing the nuts closer and closer to the kitchen door, which leads to the garden, and now he picks them up without any problem.

Well, I have a goal: I am determined to get Billy to take a nut out of my hand. I hold up a walnut for him, and he will run, from the end of the garden, eagerly towards me. But, (currently) at a distance of around one foot, he will stop, look at me and at the nut, and then turn and flee into the trees.

So I asked Enders why that was the case. It was a test. His answer came at once, without hesitation: “It’s the evolutionary thing. Squirrels that were not afraid of big animals were all eaten.”

Almost exactly one year ago I had given Enders and his brother a lesson in biology, On Evolution and Acorns, on why oak trees periodically produce bumper crops. The test was to see if he had, at the time, understood the mechanisms of evolution, and also: did he remember them? Yes, he did and knew immediately why squirrels are wary of approaching us. It is in their genes, they have learnt it in hundreds of thousands of years of evolution.

And now for the solution to the elevator problem: the man is very short, and can only reach the button for the tenth floor. When he comes home with friends, or when some neighbour happens to join him in the elevator, they will of course press the 12 for him. So simple.

If you like this kind of stuff, here are some more logic problems for you.