A simple yet difficult math problem

Arts & Sciences Science & Nature
41

Ok here’s that padlock puzzle which I just found on the Net.

[quote]
The courier problem

We need to send a valuable item to an associate in a different city. There is a very fast, very inexpensive courier service we would really like to use. Unfortunately the couriers themselves are extremely dishonest and steal everything that we put into the box. Fortunately the box has a latch which we can secure with a padlock (as shown in the picture), making it impossible for the courier to get at the contents. Unfortunately our partner in the other city does not have the keys to our padlocks. How do we send the goods safely?


Here are some points to note:

* We can only use regular padlocks, of which we and our associate have an ample supply.
* No, neither of us has 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 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 youself “Wow, that is so simple and so elegant!” you have not yet found the correct strategy.[/quote]

42

Oh, I know this one - it’s very cool. In fact, I think it was one of the concepts that led to the development of encryption systems for exchanging data over the net.

43

If I’m reading the rules right, you couldn’t just send the key first and then send the lock right? (because they would just steal the key for whatever reason) Plus I’m not thinking to myself “Wow, that is so simple and so elegant!”.

44

Since we can use the courier service as often as we want:
Send the key by courier. Send the locked box by courier. Done.

Or you double lock the box, one lock and key is with me, one lock and key is with my asscociate. Instead of directly locking the latch the two locks will be interlocked, so whichever lock you open (i.e. only one at a time) you can open the latch.

45

Oh, I know this one - it’s very cool. In fact, I think it was one of the concepts that led to the development of encryption systems for exchanging data over the net.[/quote]It is indeed

Drag your mouse over the area below for my answer.

[color=#F0F0F0]You place the goods in the box, put padlock A on it, send it to the receiver, you keep key A.
Receiver places another padlock B, on it, so it has 2 padlocks, receiver keeps key B, and returns box to sender.
Sender removes padlock A, and sends to receiver.
Receiver removes padlock B, and opens box.

The keys are never sent, and the box is always locked.[/color]<

46

BFM
Seems like a good answer.
How about putting the answer in white or small font so as not to ‘spoil’ the puzzle for others?

47

O.K. I got it. At the beginning you have a one third chance. If you could choose the other two, you would have a two thirds chance. Since the host eliminates one choice from the other two you have a two thirds chance by switching. That’s it. Simple. I may be slow but I am determined.

48

[quote]Three chaps go into a restaurant. They buy a meal for

49

The cap test

[quote]The same three chaps are being tested by a wise man.
“I have three white caps and two black caps”, he says, “You will close your eyes. I will put a hat onto each of your heads, and hide the others. You will not be able to see your own hat. You have to deduce what colour hat you are wearing.”
So he does, and the chaps open their eyes, and there is a pause, and then one of the chaps says…
[/quote]

Alright! I’ve never heard this one, and I just figured it out for myself. :slight_smile: I haven’t read any other answers yet either, so I wonder if mine varies.

I think the question is fine like this without Spack’s modification that the wise man uses just white caps. However there should be a rider that the three men, are all very smart, and they know that they are all smart.

The key is that there will always be one man wearing a white cap, and you will only be able to win if you are wearing white, so he says…

“I am wearing a white cap”.

It helps to think that they are all given a few minutes to think about it before having the caps put on their heads, and they all realise that they are smart enough to solve it in a few minutes.

So the 1st guy thinks to himself, if I see them both wearing black, I’ll say “I’m wearing white”. He also realises that if himself and one other are wearing black, the third guy will make this call. So he’s got if I see two black, I say “I’m wearing white”

So he thinks, if I see one black, and one white, and the guy with the white cap doesn’t say “I’m wearing white” then that means my cap must be white (because if it was black, he’d make the call). So now he’s also got "if I see one white, and one black, then pause to see if anyone else makes a call, if they don’t I say “I’m wearing white”.

Of course he realises that if either of the other two see one black, one white, they’ll jump to the same conclusion, pause and call “I’m wearing white”. If they don’t make this call, that means that they are seeing two white. If I am also seeing two white, that means we are all wearing white, so I should say “I’m wearing white”. So now he’s also got "if I see two white caps, wiat a bit, maybe a bit longer, then call “I’m wearing white”

Those are all the possibilites.

Brian

50

Confess Brian. You’ve been using Bayesian Statistics again.

51

This padlock one does seem impossible, and I refuse to look at the answer just yet., but I do have some questions about the rules of the puzzle.

  1. We know that the courier comapny steals anything inside the box,a nd they also steal keys right? Would they steal anything else attached to the outside of the box?

  2. Can my associate in the other city send me anything using the courier comapny too?

Brian

52

Of course you are right Brian, but can you explain exactly where the fault in logic occurs in the restaurant puzzle?

They each paid ten pounds and got one pound back = 9 pounds each (9x3=27). They gave the waiter 2 pounds. Adds up to 29.

So what error in logic is occurring here? There must be some false assumption in the above, but what is it?

53

Brian, about the padlock problem, I don’t want to give too big of a clue away but you may be making a big assumption (which I did) about the padlocks.

Look at the picture, the padlock and the clasp, to help you visualize possibilities.

[quote=“Bu Lai En”]

  1. We know that the courier comapny steals anything inside the box,a nd they also steal keys right? Would they steal anything else attached to the outside of the box? [/quote]

Ahem, read the ‘points to note’ section carefully.

The first point is also very important.

54

[quote=“Bu Lai En”]This padlock one does seem impossible, and I refuse to look at the answer just yet., but I do have some questions about the rules of the puzzle.

  1. We know that the courier comapny steals anything inside the box,a nd they also steal keys right? Would they steal anything else attached to the outside of the box?

  2. Can my associate in the other city send me anything using the courier comapny too?

Brian[/quote]

Yes and yes.

Nothing is safe unless it is inside a locked box, although you may want to consider the truth of this statement carefully.
Yes, the associate can send you something under the same conditions.

Nice thinking on the hats :slight_smile:

The logic error is of course adding the 2 to the 27 instead of subtracting it to get 25, the cost of the meal.

55

Padlock puzzle

Well, I have a solution. It doesn’t quite seem elegant, but there’s nothing in the rules against it. Just in case it is the right solution, I’ll put it in a coloured font.

[color=white]This solution relies on the latch being of the type shown in the picture.

Get your associate to send you a box locked with their own padlock (you don’t have a key to it). Don’t even try to open the box.

Get your key which has a decent sized hole in it (the hole where the keyring goes though). Get a padlock with a long thin loop. Put it through the hole in the key and thread it under the latch of the box. When you close the padlock it can not be removed from the box, except by removing the first padlock.

Send the box back to you associate. They remove the padlock, which allows them to remove (but they don’t need to open) your padlock with key attached. Now send them the goods, secured by the padlock that the key you just sent opens.[/color]

Well it may not seem elegant, but it works fine.

Brian

56

Padlock Problem:

Ouch. The answer is too easy. I gave up. I can’t udnerstand how I missed that. It’s not like I missed the slight little realisation that you might need to solve it
(you can put two keys on the padlock)
, I thought of that, but was still thinking of a too complexc answer.

Then again, my answer above still works without breakign the rules. :wink:

Brian

57

I happened to have read this problem in the book I picked up during my transit in Hong Kong on the 1st: “The Curious Incident of the Dog in the Night-Time” by Mark Haddon. Good book.

58

Bu lai En - you are correct about the hats…well done (or perhaps well done for finding the exact answer from somewhere on the net… :no-no: )

Dassgirl…correct about the money problem.

59

No way! I thought of that one myself.

Brian

60

Here’s another one for you:

The oldies are off to play bingo. It’s a rainy day, and the closest parking space they can find is a block away from the bingo hall. They only have one umbrella between them, and don’t want to trouble anyone else. Two at a time can fit under the umbrella without getting wet. They’d rather go home than get wet. Bingo starts in 17 minutes. Agatha is the slowest of the group with her walking frame. It’ll take her 10 minutes to do the walk. Bertie, with his cane, can do it in 5 minutes. Charlie can hobble along unassisted in 2 minutes. Dot, with her weekly aqua-aerobics class keeping her in shape, can do it in just 1 minute. They can only go two at a time, and one must take the umbrella back to the car each time. How can they get to the hall by the time bingo starts?

Brian