I was at a job interview and the interviewer gave me two logic puzzles. I'll repeat them:
- You are on a boat in a lake. You have, in the boat with you, a rock. You throw the rock into the water, it sinks. What happens to the water level on the shore?
- The classic "Let's Make a Deal" paradox: Suppose you're on a game show and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?
Any takers? Assuming you haven't seen them already, of course.
1. I think it would stay the same. You, the boat, and the rock are already displacing the water, so moving the rock into the water shouldn’t have any effect.
2. Yes, because if you switch, you are lowering your odds of success from 33% to 50%.
How did you do at the interview then?
1. here’s a hint: when the rock is on the boat, it displaces water via weight. when it’s in the water, it displaces water via…
2. that’s the intuitive answer. but, intuition often leads you astray, with probailities…
i gave the same answers you did. but they were both wrong…
1. i’m not sure if exerting a force on water increases volume in the same way that adding mass does. i’d say it will rise, but that’s a blind guess.
2. No. Go with 1. He knows whats behind the doors, so it is advantageous for him to offer another selection if you have chosen the car.
were you interviewing for the CIA?
woah, we commented simultaneously.
mark,
1. getting close. one more hint: the rock sinks.
2. if you haven’t picked a car already, he’s going to show you the other goat.
1. okay, so the boat (density less than water) is buoyant regardless of the rock, and only exerts a force. but because the rock’s density is higher than the water, its introduction into the lake will add both mass and density, increasing volume? I know D=M/V and V=M/D, but I don’t see that giving me a confident answer.
2. i’m not convinced that because the game-show host is baiting you with another choice, his intentions are good. if you’ve picked the goat he has no qualms about showing it to you and keeping his car. the fact that he’s giving you the option of choosing again, for me, maintains your 1 in 3 chance. just because he has eliminated a choice doesn’t mean the 2nd door has better odds. maybe it does mathematically, but a car is not an X and a goat is not a Y. Cars cost sponsors money. You can’t just ask a guy who has picked the car to go home happy. So by giving him a chance to change his mind, you are in my mind already defeated by his choice, and looking for a way out. If the game show host is somehow impartial, then I guess you could switch your choice and rely on the probability of 2/3 rather than 1/3. but nobody is impartial. especially you, otherwise you’d be happy with the goat.
* when the rock is in the boat, it displaces a volume of water that weighs the same as the rock. (objects displace their own weight, when floating)
* when it’s in the water, it displaces a volume of water equal to its own volume.
* because it sinks, we know it’s more dense than water.
* therefore we also know the rock has a smaller volume than what it would take for the equivalent weight in water (higher density, lower volume needed for equivalent weight)
* so, the rock displaces less water in the water than it did in the boat.
* so, the water level on shore… goes down.
not the kind of question you’d expect during a phone-screen.
the door one is much trickier. this is a pretty good explanation. bottom line: you should always switch.
and the explanations, even when i can follow them, never sit right with my intuition; they make sense mathematically, but never click in reality – but i always sucked at probabilities anyway.
1. the rock explanation is sufficient, and i will remember it always.
2. monty hall is an ass. the “solution” to his “problem” is highly unsatisfactory, and sounds a lot like Intelligent Design.
1. Tricky! I never would have thought the water level went *down*.
2. Oops, I meant to say that you are raising your probability of success from 33% to 50% if you switch. I had heard this one before though. </me=cheater>
Thanks though, that was pretty interesting :)