Martyr to the cause of attractive consumer electronics.
Tuesday, November 27, 2012
Monday, November 5, 2012
Labyrinth Puzzle
I was thinking the other day about that Labyrinth puzzle... that scene where Sarah runs into a pair of muppet-like creatures guarding two doors. The premise, according to one of the creatures, is that one of the guards always tells the truth, and one always lies. And if Sarah chooses the correct door, she gets a cheesecake, and if she chooses the wrong door, she dies. So all she needs to do, presumably, is figure out which guard is telling the truth, and ask him which door to take. So I was thinking, what is the best approach to this situation? Here's one approach which I thought of: Ask second guard whether he agrees with the first guard, specifically that one guard always tells the truth and the other always lies. His possible answers are:
1) Yes, one tells the truth and one lies
2) No, they both always lie
3) No, they both always tell the truth
4) No, one sometimes tells the truth and the other always lies
5) No, one sometimes tells the truth and the other always tells the truth
6) No, they both sometimes tell the truth
If he answers 1), we know they are both lying. The reason is that if the statement "one tells lies one tells truth" were true, one guard would necessarily contradict the other. So this means either that both guards always lie (case 2), one guard always lies and the other only sometimes lies (case 4), or that they both sometimes lie (case 6). If the second guard answers 2) he must be lying. The reason is that if it were true, the second guard couldn't say it because he must lie. So this means one of two things: the second guard always lies and the first guard sometimes or always tells the truth; or, the second guard sometimes lies, which means the first guard must sometimes or always lie (since the first guard just said that "one always lies and one always tells the truth."). If the second guard answers 3) he must also be lying. The reason is that if it were true, the first guard couldn't contradict him. So in this situation we draw the same conclusion as if he had answered 2). If the second guard had answered 4) he could be lying or telling the truth. If he's telling the truth, we know that the first guard always lies, since the second one isn't lying (by assumption). Or, if the second guard is lying, the situation is consistent with 1)---which means the first guard always tells the truth and the second always lies---2), 5)---which means that the first guard always tells the truth and the second sometimes tells the truth---and 6). If the second guard says 5), he can again be telling the truth or lying. If he is telling the truth, then he always tells the truth since we know that the first guard at least sometimes lies because he contradicts the second guard. If the second guard is lying, the situation is consistent with 1)---in which case the first guard always tells the truth and the second always lies---2), 4) or 6). If the second guard answers 6), he could be telling the truth or lying, and if the latter the situation is consistent with 1)---in which case he always lies---2), 4), or 5)---in which case the first guard always tells the truth. Let's summarize these results in a table with ordered pairs (,), the first entry of the pair refers to the first guard and the second entry to the second guard. T means he always tells the truth, L means he always lies, and U means you can't trust him either way. So depending on how the second guard answers the question, we have the following possibilities:
1) (L,L) or (U,L) or (L,U) or (U,U)
2) (T,L) or (U,L) or (L,U) or (U,U)
3) (T,L) or (U,L) or (L,U) or (U,U)
4) (L,T) or (T,L) or (L,L) or (T,U) or (U,U)
5) (U,T) or (T,L) or (L,L) or (U,L) or (L,U) or (U,U)
6) (T,L) or (L,L) or (L,U) or (U,L) or (T,U) or (U,U)
So actually we made a lot of progress: Out of the 9 possible combinations of trustworthiness between the two guards, we have eliminated all but 4 or, at worst, 6 scenarios.The problem is that in all cases, it's possible that both guards sometimes lie. I don't see question you could ask to eliminate this possibility. So the upshot is that you can't trust anything either one of them can says as true or false. So you might as well ignore them and pick a door.
1) Yes, one tells the truth and one lies
2) No, they both always lie
3) No, they both always tell the truth
4) No, one sometimes tells the truth and the other always lies
5) No, one sometimes tells the truth and the other always tells the truth
6) No, they both sometimes tell the truth
If he answers 1), we know they are both lying. The reason is that if the statement "one tells lies one tells truth" were true, one guard would necessarily contradict the other. So this means either that both guards always lie (case 2), one guard always lies and the other only sometimes lies (case 4), or that they both sometimes lie (case 6). If the second guard answers 2) he must be lying. The reason is that if it were true, the second guard couldn't say it because he must lie. So this means one of two things: the second guard always lies and the first guard sometimes or always tells the truth; or, the second guard sometimes lies, which means the first guard must sometimes or always lie (since the first guard just said that "one always lies and one always tells the truth."). If the second guard answers 3) he must also be lying. The reason is that if it were true, the first guard couldn't contradict him. So in this situation we draw the same conclusion as if he had answered 2). If the second guard had answered 4) he could be lying or telling the truth. If he's telling the truth, we know that the first guard always lies, since the second one isn't lying (by assumption). Or, if the second guard is lying, the situation is consistent with 1)---which means the first guard always tells the truth and the second always lies---2), 5)---which means that the first guard always tells the truth and the second sometimes tells the truth---and 6). If the second guard says 5), he can again be telling the truth or lying. If he is telling the truth, then he always tells the truth since we know that the first guard at least sometimes lies because he contradicts the second guard. If the second guard is lying, the situation is consistent with 1)---in which case the first guard always tells the truth and the second always lies---2), 4) or 6). If the second guard answers 6), he could be telling the truth or lying, and if the latter the situation is consistent with 1)---in which case he always lies---2), 4), or 5)---in which case the first guard always tells the truth. Let's summarize these results in a table with ordered pairs (,), the first entry of the pair refers to the first guard and the second entry to the second guard. T means he always tells the truth, L means he always lies, and U means you can't trust him either way. So depending on how the second guard answers the question, we have the following possibilities:
1) (L,L) or (U,L) or (L,U) or (U,U)
2) (T,L) or (U,L) or (L,U) or (U,U)
3) (T,L) or (U,L) or (L,U) or (U,U)
4) (L,T) or (T,L) or (L,L) or (T,U) or (U,U)
5) (U,T) or (T,L) or (L,L) or (U,L) or (L,U) or (U,U)
6) (T,L) or (L,L) or (L,U) or (U,L) or (T,U) or (U,U)
So actually we made a lot of progress: Out of the 9 possible combinations of trustworthiness between the two guards, we have eliminated all but 4 or, at worst, 6 scenarios.The problem is that in all cases, it's possible that both guards sometimes lie. I don't see question you could ask to eliminate this possibility. So the upshot is that you can't trust anything either one of them can says as true or false. So you might as well ignore them and pick a door.
Thursday, October 25, 2012
Sunday, August 26, 2012
Tuesday, August 21, 2012
Upon vs. Apon
I can never remember how to spell this word. I'm always getting dinged by spell checker. Turns out the correct spelling is "upon." Let's see if I remember this tomorrow. It turns out that in Middle English the spelling was "apon," which may be where the confusion started in my childhood. I actually think we should write apon, otherwise
we should change the spelling of a million other words for consistency. Like, we write
"a lot" rather than "u lot." Or "agressive" rather than "ugressive." Or "anonymous" rather than "unonymous." Or "attractive" rather than "uttractive." Or "anderstanding"
rather than "understanding."
Saturday, August 11, 2012
Fascist Duck
My visit to Munich got me thinking about Hitler and what happened in WWII. Its pretty hard for me to understand what happened. Found this funny Donald Duck cartoon from 1943 on youtube, found it kind of surreal and touching.
Tuesday, July 31, 2012
Floors
Here in munchin' I've been taking the elevator up and down the building a lot, which got me thinking about floors. So let me ask the reader a simple question: If you walk into a building, what is the first floor you encounter? We USers answer this question in an exceedingly elegant way: the first floor you encounter is the first floor. Brits call it the ground floor, Czechs call it the prizemi floor, and apparently the Germans call it the E floor (not sure what the E stands for). I think what the Euronids have in mind is that these are really names for the zeroth floor, but don't want to admit it because no normal person starts counting from zero. However, one cool thing about the Euro system is that the basement and lower levels can be called the -1,-2 floors etc. In the US system, I guess we should therefore call the basement the zeroth floor. But this seems unsatisfactory, since 0 is too important a number to be applied to the basement, a place where people only go to store items which really should have been thrown away. And 0 doesn't really give the feeling of being below something, its kind of at the center. And we can't exactly skip over 0 and call the basement the -1st floor either. Because then everyone would ask what happened to the zeroth floor between the first floor and the basement. In the US I think we have failed to find an adequate solution to this problem.
In parking garages we call the lower levels the "A level" or "B level," as though we need an exotic new number system to count floors below the ground. What does the reader think we should do?
Tuesday, June 5, 2012
Saturday, May 26, 2012
Friday, February 17, 2012
Thursday, February 16, 2012
Tuesday, February 14, 2012
Cashing In
Given the popularity of this blog, I thought it was about time to turn this thing into a money-making machine. So I signed up to post ads in the right hand column of the page. Please feel free to click, and let the cash flow into my pocket. I already see an add for some wireless thingy over there. Hopefully more will come soon, and my blog will be the go-to place for all your online shopping.
Saturday, February 11, 2012
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