Friday, November 2, 2012
Logic--Huh! Good God! What is it good for?
Setting aside ontological arguments for the moment, let's begin to sharpen our logic skills. For Tuesday, please tackle these logic puzzles.
Work patiently and with a pencil, reasoning out the consequences of
each statement and, where appropriate, its speaker. In your post
examine the ways your thinking changed or developed to accommodate this
task. What was most difficult? How did you arrive at the answers? If you get angry at logic, take a break and read this.
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During the first few puzzles that I started solving my brain was anxious to solve them and figure it out. As I continued on with each puzzle my brain started to cringe at the idea of doing another. It became easily distracted and flustered. I wanted to give up and be done with it. The most difficult part was the last bits of the puzzles where I was almost at an answer but I was being held back either by a small piece wordings or an unknown object. It was difficult to keep on task after my mind had decided that it was done. To arrive at the answers I tried several different methods of solving, including guessing, but then I realized that the only way it can be done is by doing out the work because there were no short cuts. The last puzzle that I did was The Probem of the Light Switch, at first this problem seemed simple, just find out wether the light was on or off at noon. The work to be done for this problem was easy at first and then it got more difficult because it was hard to keep the numbers straight. When it came down to the last 10 numbers to divide by 2, the time became 11:59.98535157 am and seemed as tough it was never going to end. My mind was becoming distracted by everything to avoid solving this but when I completed the problem I felt accomplished.
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ReplyDeleteIn the process of solving these puzzles, I set up hypothesis, and the tried to prove it by showing the evidence that could disprove it. I think the difficult part of these parts is that I have to be very organized with my logic, and cannot let my thoughts jump around. Soon I found out that I got used to one answer to the puzzle at first, and later I realize that the puzzle could be solved in another way. Because my thinking had restricted by my previous conclusion, I tended to spend more time trying to disprove my second conclusion. It was hard to break the bondage of my first thought, but once I achieved that, my mind was ready to open up to more answers.
ReplyDeleteWhen I started to solve the first puzzle, I set up a hypothesis that B was a knight at first, which meant that "A said that there is exactly one knight among [them]” was true. If there was only one knight, and B was the knight, A and C must be knaves. However, if A was a knave, his/her statement must be false, so that what B said must be wrong, too, which meant that it was impossible. Then, I supposed that B must be a knave. If B was a knave, his/her statement would be “A didn’t say there is exactly one knight among [them]. This could be interpreted in two ways: one was that A didn’t talk about knights; the other was that there wasn’t exactly one knight among them, which meant that there could be no or two or three knights among them. If it was the first interpretation, C could be either a knight or a knave because I didn’t know whether A talked about knights or not. If it was the second interpretation, there were three hypotheses (no, two, three knights). However, the three knights hypothesis didn’t exit because if B was a knight, C must be a knave which was demonstrated in the very first hypothesis. If there was no knights which meant that all of them were knaves, then B meant there were not exactly one knight, and C supported his/her statement. In this case, the logic made sense. Also, if there were two knights among them, and I already assumed that B was a knave, which meant that A and C were the two knights. In this case, C didn’t support what B said, which also made sense because I didn’t know what A said. So far, I figured out more than one conclusions for this puzzle, and they all seemed logical to me.
Through the process of solving this puzzle, I got confused several times, and didn’t know what exactly I was proving. My mind became more organized as I kept going, but I still wasn’t sure about my conclusion. I think it is like a tree diagram. Although there are so many possibilities, I need to keep track of where I am going. It is the hard part, but at the same time, it is the only way that can lead me to the conclusion.
**PS: I find “The Super Bullet” very interesting because there is an ancient Chinese fable that tells a similar story. A man sold both spears(矛) and shields(盾). He flaunted his shields, “My shields are the solidest shields ever, and nothing sharp can penetrate my shields.” Then, he showed off his spears, and said, “My spears are the sharpest in the world, and can penetrate everything hard.” After that, one person asked him, “Could you please use your spear to penetrate your shield, and see what will happen?” The seller didn’t know what to answer. This fable was excerpted from a book called Hanfeizi written by Hanfeizi in around 250 B.C. The significance of this fable is now used as a Chines idiom called 自相矛盾(zi xiang mao dun) which means that what a person says is contradictory to his/her own statement (paradox). : )
I have come to the conclusion that I do not like to solve puzzles like this because they are very overwhelming and frustrating. I have attempted to do these types of problems before and I’d given up very quickly because I got impatient, so when I started to tackle these, I told myself to give them a chance. Starting each one was the most difficult because there was way too much information all at once. I needed to write down on a piece of paper the different scenarios, and go from there.
ReplyDeleteThe first scenario was the knight and the knave puzzle. I wrote down on a piece of paper the different possibilities and came to the conclusion that there is more than way answer or scenario because A,B, or C could be a knave meaning they are lying so each scenario will be switched around. I couldn’t come up with one answer because each time I would question who was who, and realized it couldn’t just be that one. The second part of the puzzle I came the conclusion that the inhabitant is just and knave and is lying, because it is a fact that 2+2=4 not 5.
The super bullet question is very interesting, and I think I have heard ones that are similar before. I think that what would happen is that both the bullet and the armor will meet at force however they will just stop once they touch.
Another puzzle I looked at was the on and off light switch. I wrote down all of the times the light would be switched until it was a very very very small number. I then counted backwards from two minutes until the small number and it is how I reached my answer that the light switch will be on.
Solving these problems were very mind boggling, and I can’t say I had fun, but they were a good experience to get my mind really in motion. It got frustrating but I knew it had to get done so I pushed on through.
On the Island of Knights and Knaves
ReplyDeleteThe first logic puzzle I attempted to tackle was the Knight and Knave one. I found this quite simple once I looked at the puzzle logically and worked out the answer neatly. B makes a claim: “A said that there is exactly one knight among us.” B’s statement can either be true or false. I started with the possibility that B’s statement was true. If B is truthful, then C is a liar. If B is truthful, then A really did make the claim that there was one knight among them. A can’t be truthful because then there would be two knights, making A’s statement false. Therefore, A is a liar. If A is a liar, then there isn’t exactly one knight among them; there are two, three or none. None of these possibilities work, so I concluded that B’s statement was in fact false. Now, if B’s statement is false, then C is truthful. If B is a liar, then A did not claim that there was exactly one knight (A could have said anything else). I concluded that B is a knave and C is a knight.
For the second part of the question, I came to the conclusion that the person speaking is lying, and therefore a knave. He gave the reader two possibilities to choose from; either he is a knave, or two plus two is five. If I chose the first possibility, it would result in a paradox because it would mean that he was telling the truth about being a liar which would actually make him truthful except that the truth is actually that he is a liar. Therefore, the second possibility must be chosen. However, two and two does not equal five and that led me to the conclusion that the speaker is a liar.
My thinking didn’t really change while I was solving these puzzles because the solution was quite easy to arrive to once I had organized all of my thoughts and written them down on paper. I already work systematically with logic puzzles.
The Surprise Test
The Students’ reasoning went wrong when they concluded that the test couldn’t be on each of the days preceding Friday because of the chain reaction it caused when Friday was dubbed as a “no test” day. On Wednesday, the test could be on Thursday or Friday. On Tuesday, the test could be on Wednesday, Thursday or Friday. On Monday, the test could be on Tuesday, Wednesday, Thursday or Friday. It is only after the day has passed that the possibility of it being a quiz day is ruled out.
This problem was a little bit more difficult for me to solve because the problem states the issue in absolutes and explains the reasoning behind it. Even though the reasoning and the absolutes were incorrect, it was still difficult for me to distance myself from the problem and look at the facts without letting myself get sucked in by the seemingly logical proof.
The Problem of the Light Switch
I was not able to solve this problem. Or in fact, the solution is that it will never be noon. What I concluded after reading the puzzle was that this was an exponential problem with a solution that approaches infinity. If the solution approaches infinity (the light switch must be turned on and off an infinity number of times before it is noon) then it can never actually be noon because that means that we were successful in turning the switch on and off an infinity number of times, which is actually impossible to do.
Also, since we don’t know the smallest unit of time or the specific time at which the switch is flipped for the last time, we cannot know whether or not the light was on or off at the end of infinity.
The Super Bullet
ReplyDeleteThis was my favorite puzzle, and in my opinion, the most difficult one because it required me to accept an answer which does not seem realistic and cannot actually exist in real life. The bullet penetrates anything it hits and nothing can penetrate the plate. This cannot exist in real life. In real life, one of the objects will emerge as the victor; one of them must be stronger than the other, even if it is by 0.0000000000001%. Therefore, the bullet and the plate can’t actually come into contact. If they did, then it would be a paradox since they are both of equal strength. However, common sense doesn’t let me accept that no matter how many times I shoot, the bullet will not come in contact with the plate. That is why I had to shut off common sense; something a lot easier said than done.
The Monkey
Honestly, this one was a pain. I can’t even count the number of times I lost track and had to start all over again. I got so wound up in all of the ups and down that I got frustrated and had to take a break. It didn’t really do me any good. I couldn’t solve it after I came back either. I was able to find the term for the younger monkey’s current age but I was too confused to find a term for the mother’s age. This problem requires the optimum organizational skills. However, I do understand that I have to work backwards and carefully comb through it in order to solve it. I need to look more at the bigger picture and fully understand what the puzzle is trying to tell me before I try to break it down and tackle it piece by piece.
When the second question of On the Island of Knights and knaves asked what I could conclude, I thought to myself: knaves lie so it would not tell the truth and say it was a knave. Two plus two does not equal five, but knaves do not tell the truth. Therefore, the person cannot be from the island.
ReplyDeleteIn The Surprise Test, the students went wrong because even though their first thinking was correct which was that if the test was not given by Thursday, it must be on Friday and therefore it will not be a surprise when the teacher gives them the test. Although, the students’ logic became flawed when they started to rule out Thursday then Wednesday and so on, finally concluding the test would not be given. This is because yes, the test would not be a surprise if Friday is the only day of class left of the week, but if it is Wednesday there are still two more days, providing more than one option. This is where their reasoning went wrong.
In The Problem of the Light Switch I considered the fact that no living being with opposable fingers on earth can move them quickly enough to accomplish turning the light on and off cutting the time between each flipping in half. Also, it said there would be an infinite series of flips. Infinite means never ending. Therefore, one cannot conclude whether the light will be on or off at noon given the fact that the flipper will never stop flipping. Poor thing.
The process I went through systematically was this:
look at question
look at key facts or words
create reasonable possibilities
consider which possibility is most logical given the provided information and circumstances
see if my answer is correct
The most difficult task was coming up with as many possibilities as I could. Many times, I believe I’ve come up with them all, but in fact I have not, and this applies to puzzles outside of Tuesday’s assignment.
Puzzle 1-
ReplyDeleteWhen the stranger asked B what A said I knew that either he could lie about what A said or tell the truth. If B was telling the truth, than it would be true that there was only one knight among them. This could not be true because if A had told the truth than he/she was the knight and B was the knave. When C said that B was lying I knew that either B or C was lying, and that both B and C could not be knights or knaves. So, I concluded that B was a knave and C was a knight.
Puzzle 2-
First I conclude that Raymond Smullyan is a very confused and distrustful man and I would never want to be his friend because I feel like he would always question my truthfulness.
I concluded that the inhabitant cannot be a knight or a knave. There were only two possibilities. One is that the inhabitant was a knight. This is not true because 2+2≠5. If the inhabitant was a knave then by him/her saying that he/she was a knave would be the truth and knaves always lie.
Puzzle 3-
At first I thought “how could it be a surprise test if the students came to class everyday expecting a test?”. Well, then I read their reasoning. I agree that with the students reasoning that the test could not be on Friday because with the test not being on Thursday, Friday became the last possible day which again would not be a surprise. The students reasoning went wrong when they concluded that the test was not going to happen. The teacher told them they were going to have a test next week and that was the only sure statement the teacher gave. So, the students could have expected a test on any day and knew for sure on Thursday after class that the test would be on Friday.
Puzzle 4-
First I thought about how annoying lights like these are. And then I wrote out each time increment and realized that what the puzzle meant by infinite was that the time could never reach noon. So, this puzzle gives no way to answer if the light switch will be on or off at noon. It tells you whether the light will either be on or off BEFORE noon but by cutting the time in half we will never be able to reach noon.
Puzzle 5-
At first I was not sure what would happen if the bullet HIT the plate. When the question asked what would happen if the bullet was SHOT at the armor I knew the bullet would go through the armor. The flaw is that the puzzle never specified under what circumstances the bullet and armor plate were impenetrable.
Puzzle 6-
The first piece of knowledge I got was that the pulley was weighted on either side by a weight and a monkey. The rope that is between the two, weighs 4 ounces for every foot of rope. The monkey and its mother’s ages equal 8 years. So the age of the mother could be 6, 5, or 4 because the mother must be older than the baby. “The mother is twice as old as the monkey was when the mother was half as old as the monkey will be” this part of the problem contradicts itself so; the age of the mother is still twice as old as the monkey. The mother is also three times as old as the monkey. Together, the mother is 5 times as old as the monkey. The weight and the monkey weigh the same amount. I’m lost.
Errors in the title of the page:
ReplyDeleteAfter staring the the header for an unreasonably long time, I decided to read it aloud to myself. Oh wait! There are two 'the's in the title! I found one error. But as for the second one, no such luck. I can't find another error! This seriously frustrated me and probably contributed to the frustration I encountered when trying to solve the other ones.
I'm trying to think of a synonym for frustrated which really captures the feeling I had trying to solve these, but no other words really work so I'm just going to use frustrated again. It was frustrating working through these puzzles because I lost track of my thinking and sometimes, even though I know these are LOGIC puzzles, the puzzles seemed a bit illogical to me. Would a teacher really go to such painstaking measures to make sure that the students had absolutely no clue as to when the test was? Maybe I just didn't get into the spirit of solving logic puzzles but this is how my thinking changed. I think the students reasoning went wrong because they took so literally what their teacher said to be about the test being a surprise. Maybe the teacher just knew his students really well and knew that they would conclude that their would be no test and he took advantage of that and gave them a test on Wednesday. Isn't that logical? Basically, towards the end and even the middle I wasn't sure what counted as logical anymore.