It takes a nation to protect the nation
Does anybody have their favourite puzzles and equations? Those are ones which in some way are very elegant or cute. Here are a few of mine.
1. The Two Guards (a 2000-year-old classic)
What I like about this is it makes me think of Socrates teaching the first rules of reason, and that effectively, 2000 years ago, people were doing boolean algebra without realising it.
You stand at a fork in the road. Next to each of the two forks, there stands a guard. You know the following things: 1. One path leads to Paradise, the other to Death. From where you stand, you cannot distinguish between the two paths. Worse, once you start down a path, you cannot turn back. 2. One of the two guards always tells the truth. The other guard always lies. Unfortunately, it is impossible for you to distinguish between the two guards.
You have permission to ask one guard one question to ascertain which path leads to Paradise. Remember that you do not know which guard you're asking -- the truth-teller or the liar -- and that this single question determines whether you live or die. The question is: What one question asked of one guard guarantees that you are led onto the path to Paradise, regardless of which guard you happen to ask?
2. Wiring a 2-Way Lighting Circuit
If you are wiring a 1-way lighting circuit, its pretty obvious, but how do you wire a lamp so that either of 2 switches can operate it? I think the solution is cute, and notice how it relates to the previous problem.
3. Boatman with Fox, Chicken and Grain
This is a children's puzzle, but I like the fact that the solution and undoes itself at one point (as obviously has to happen in the Towers of Hanoi problem).
A man has to get a fox, a chicken, and a sack of corn across a river.
He has a rowboat, and it can only carry him and one other thing.
If the fox and the chicken are left together, the fox will eat the chicken.
If the chicken and the corn is left together, the chicken will eat the corn.
How does the man do it?
4. The Sum-Product Problem
There are many variants to this problem, but this will do for starters. What's nice about it is the way the absence of knowledge, and 'no reply', may themselves be used to make fine deductions. Its as if the information is pulled out of thin air. Please note that some versions of the problem give an invalid upper limit of 20 for the numbers - which makes the problem impossible.
Two numbers (not necessarily different) are chosen from the range of positive integers greater than 1 and not greater than 100. Only the sum of the two numbers is given to mathematician S. Only the product of the two is given to mathematician P.
On the telephone S says to P, "I see no way you can determine my sum."
An hour later P calls him back to say, "I know your sum."
Later S calls P again to report, "Now I know your product."
What are the two numbers?
The 2nd solution on this webpage, is a nice one:
If you like that one, here's an easier but similar variant.
An old king is about to die and he has no offspring to inherit the crown. So he summons the three wisest men from his kingdom and puts them to a test. He tells them that he is about to put them in a room and have his aide put a hat on each of them. Each hat may or may not have a dot on it, but at least one hat will have a dot. They may not touch the hats, nor communicate in any way. The first one that correctly identifies whether his hat has a dot will become the next king. If he is wrong, or if he breaks the rules, he will be killed. Then he sends all three wise men into the room.
The king then tells his aide to put dotted hats on all three.
A few minutes later one of the wise men returns and announces proudly that he has a dot. How did he know?
5. Favourite Equations
Euler's identity is nice, because you would think it impossible that two irrational numbers and a complex one, be related in such a simple way:
e^(i pi) = -1
The correction for E=Mc^2 is good as well, because the original equation feels just a little bit too simple for describing what is happening near the speed of light.
E^2 = (mc^2) ^2 + (pc)^2
6. Twelve Coins
Allegedly posed to a female Israeli soldier by her male subordinates to decide whether they would "respect her" -- she solved it, and they did.
You have twelve coins, eleven identical and one different. You do not know whether the "odd" coin is lighter or heavier than the others. Someone gives you a balance and three chances to use it. The question is: How can you make just three weighings on the balance and find out not only which coin is the "odd" coin, but also whether it's heavier or lighter?
The solutions I've seen are long and inelegant. I'll put mine if anyone's interested.
Hint: we've reverted back to problems 1 and 2 again :-) What I like about this one is that, not a single iota of information is wasted in the computation of the solution; it employs the maximum variation possible, so is never boring.
7. Where's The Missing Dollar?
You see a shirt for £97, you can't afford it so u borrow £50 from your mum and £50 from your dad which equals £100. You buy the shirt and get £3 change.You give your dad £1 and your mum £1 back and you keep the other pound. So now you owe your mum £49 and dad £49, 49+49 =£98, plus your £1 =£99. Where's the missing £1?
That's what happens when you mix up assets and liabilities, and gaily add them together. Its a bit like add kg weight to km distance.
What's interesting about this is that it becomes clear that you are dealing with 3 different dimension:
Now suppose that this problem represents the entire country, (yes, its a very poor country, it only has 100 notes of £1, and 1 shirt, and everyone else is naked). And suppose that the shirt costs £100 and I borrow £100 off my parents to buy it.
Then suppose that the government prints another 100 notes of £1 and gives it to you, their agent. They have then stolen from the rest of us to give value to you. Then those total £200 of notes will chase the same amount of goods.
You now buy the shirt off me for £100, and I repay my parents. We are then left with: my parents have £100 notes, I have nothing just like when I started, the shopkeeper has £100 notes, and you have the shirt.
Seeing that the total value of the assetts of the nation is £200, you deduce that the shirt should now sell for £200. Now none of us has enough money to buy the shirt. By inflating the currency through printing more notes, the government has made us all poorer. And by giving you money through the printing press, it has re-distributed wealth from all us to you.