The summary of the game of dividing mungsFive prisoners, in the order of 1-5, respectively, grab mung in a sack containing 100 mungs. Each prisoner is required to grab at least one mung. The person who catches the most and the least will be executed, and the person who catches the same number will also be executed. Moreover, they cannot communicate with each other, but when catching, they can touch the remaining number of mungs.Background They are all very smart people. Their principle is toask for life first. 100 do not have to be divided.My Analysis*My perspective is based on the fact that everyone knowsabout others’ level of intelligence. Each person takes a plan, so that when the rest of the people take the best plan,their own survival probability is the highest. If there are multiple schemes to maximize their survival probability and they arethe same, the scheme to kill the most people will be adopted.If my hypothesis is obtainable andright in some ways, then it may lead to a very horrible ending.We defined m_n as the number of mungs the nth people take, and M_n as the sum of mungs taken by the top n people.Lemma 1When n people(1 ≤ n ≤ 3) has taken mungs, if the number of mungs taken by them make 20n < M_n < 95 + n, then the (n+1)th people should take mn+1=min(96 - Mn + n , [(M - 1) / n]).Prove of Lemma 1In this scheme, you can ensure that you will not die. At the same time, people who have not taken beans are the most likely to die. 96 - M_n + n is to guarantee that the last left people can have at least one mung to take, and the one who takes the mungs before the last one will take at least 2 mungs. [(M - 1) / n] is to ensure that you get at least 1 less mungthan the person who gets the most.Due to the fact that Mn > 20n, we have [(M - 1) / n] ≥ 20. This not only ensures that the number of beans you take is not the most, but also that no one else can get so many beans, so you can survive. If [(M - 1) / n] < 96 - Mn - n, he kept the number of beans left to a minimum while ensuring his own survival, so that more people could be killed. If [(M - 1) / n] < 96 - Mn - n, he can only take one bean for each of the remaining people to ensure his own survival and kill all the remaining people.Corollary 1If the first person wants to survive,he can’t take more than 20 beans. Otherwise, if the latter person takes lemma1, he will ensure his own survival. At this time, the first person will die because he takes the greenest beans, and the last 1-3 people (according to the number of green beans taken by the first person) will die because he takes the least number of beans.Lemma 2When n=2 or 3, if Mn ≤ 20n, then the (n+1)th people will take mn+1 = [Mn / n + 0.5] mungs to ensure his survival possibility.([Mn / n + 0.5] is the rounding of the mean)Because if and only if in thiscase, as long as the difference between the maximum and minimum number of mungbeans taken by the front person is not less than 2, they can ensure their ownsurvival. (Otherwise, the survival range will be narrowed)For the fifth person, this condition may not be true. For example, seeing that the first four people took 62, maybe 14+16+16, or 15+15+15+17 ,so whether he took 15 or 16, he had a chance but could not ensure his survival.And the difference between the maximum and minimum value of mungs taken by all people is not greater than 1, and all people have to die.Lemma 3When everyone is extremely selfish, the first two people are not likely to survive.This is because by lemma 2, if the third to fifth people will take the plan of survival probability for them, if the difference between the number of mungs taken by the second and the first people is more than one, then the first two people have the maximum and minimumvalue, and must die, if the difference is not more than one, then all people must die.ResultsSince the first person has no survival probability, his goal is intriguing:If you don’t have a chance to survive—Option 1: kill as manypeople as possibleOption 2: save as manypeople as possibleAccording to my hypothesis, it should be the former. Since the first person doesn’t havea chance of survival, let’s all die - Take 96 mungs!But if the first person is a little compassionate and choose 2. Then, he will take 21-33 beans. According to lemma1, the 2nd-4th people will survive.Therefore, according to the different understanding of the question, there are two solutions: 1. All people will die. 2. The 2nd to 4th people will survive.For the first prisoner, he will face a philosophical dilemma:If you can’t live, will you chooseto let others bury you or let others live well?The further analysis is in the part “Moral Analysis”.Theoretical AnalysisI have to point out: because there is no “everyone knows that other people are also very smart” condition in the title, there will not be 96 beans selected by A.Let’s see a very basic question that there are only 3 people ABC, 10 mungs, and all other condition is the same.At first, B is very nervous, he began to think.For him, there are 3 tactics:Better : He himself alive.Medium : All the people die.Worse : He is dead, but someone else is alive.Then he began to predict A’s behavior:If A takes 8 beans, B takes 1, C takes 1. All dead.If A takes seven beans, now it’s B’s turn. If B takes 1 bean, C dares not take 1, and must take 2. C survives and the rest die. If B takes two beans, C can only take one. B lives alone. Because B is a rational person, B will take two. A and C are dead.If A takes six, B takes three; if A takes five, B takes four. It’s all B’s and both A and C are dead.B has found the rule, that is, let 9 the number of his own take between AC, he can guarantee life.If A takes four, now it’s B’s turn. If B takes five, C can only take one. A live alone and BC dies. If B takes four, C will die with three people no matter how many beans he takes. If B takes three, C chooses to take three when he knows that the first two peopletake seven, and the three people die together. If B takes two or one, C will choose to take three, and C will live alone. A and B died. B surprised to find that no matter how he chooses, he will die. He won’t choose to let C live. So, B chose to take three.If A takes three, B thinks a little, and chooses three people to die together.If A takes two, B will take three, but C laughs. He doesn’t take five, nor four, nor one. He takes three. All three died together.If A takes one, B will choose one, all three dead. If B chooses 2, C will choose three dead. If B selects 3 or more, and C select the average number of A and B. A and C will live, but B will die, so B won’t make this choice.But A also thought about the wholeprocess. A discovered a sad fact. If B is very smart, no matter how he chooses,he will die. In this case, A places his hope on that B is not very clever. Hesmiles and chooses four beans. Then A, B, C all dead.Then We conclude a theorem: If three people have n beans, n > 3, and A doesn’t know whether B and C are rational, he can choose [n / 3]. If all three ABC people are rational, they will die together. Then if the D and F participate in the game, I myself wrote a code in C++ to analyze all the possibility. It is the same.(See Annex)Let’s see the result:If all of them are rational: 100 left for AA will chose 1090 left for BB will choose 1179 left for CC will choose 1168 left for DD will choose 1058 left for EE will choose 10 This based on that they are abusolutely clever.And obviously, they all dead. Moral AnalysisWe will find that among all the possibilities A’s life depends on B.In other words, if A is smart enough, he will think that his life and death may be decided by B. For example: A grabs 5, B has a way to make A live (grabs 90).But, this kind of decision, needs a premise, name: B has the gratitude mentality.Let’s define the gratitude mentality: Weak gratitude mentality: If others show good tome, on the premise of not affecting self-interest, I choose to be good to him. Strong gratitude mentality: If other people choosenot to be bad to me when they can be bad to me, on the premise of not affecting self-interest, if I can be bad to him or not, then I choose not to be bad to him.As the gratitude mentality is real in the world, A has a great chance of survival.As long as A doesn’t kill B, andlets B go. Although B knows that he will die, A will live as long as B has astrong gratitude mentality.But in the world, the weak gratitude mentality is relatively common, and the strong gratitude mentality is relatively less. ——If I live, let me be good to you. If I dead, I don’t care if I’m good to you.If A is a mediocre but not despicable person (only seeking self-interest, not seeking harm), then his successor B will enjoy the dividend of the forerunner when he has a strong gratitude mentality. Otherwise, A will be the first to die.In other words, A’s survivaldepends on B’s strong gratitude mentality. Whether BCDE must die or not dependson whether A pursues harm.Therefore, in the real social model(under the assumption of self-interest but not necessarily at the expense ofothers), A will not choose to catch 96 and let everyone die.However, no matter what happens, Bwill die. It’s inevitableSo, let’s suppose that A is an elevated person. Also, let’s define the nobility first.Weak noble:If you can be self-interest, self-interest. If you can’t be self-interest, it’s better to benefit others.(this definition is not strict, because sometimes self-interest involvesharming others. The strict definition is too complex, so it is omitted. Inaddition, gratitude mentality is also a specific example of weak and noble.)Strong noble:There is no difference betweenbenefiting people and self-interest.Strong and noble rarely exist inthe world, generally only in close relatives or people with religious beliefs. Weak high is relatively common.Assuming that a is a weak and nobleman, he realized that he could not escape death without other noble people inthe world. Since both sides are dead, it’s better to be a noble person.So, A choose to catch 1. This means thata has the least chance of sacrificing others by himself. If B is vulgar, he will choose two. C, Dand E are all vulgar people, only two of them should be arrested. (1, 2, 2, 2,2) all dead. A willing to save the world, but failed. But as long as there is one person in B, C, D and E who is willing to catch 50, he can save all peopleexcept himself and A. A’s death is to be benevolent. His own death is tosacrifice his life for righteousness. Because two people choose to sacrifice,others can be saved.At last, I’d like to quote Ksitigarbha Bodhisattva’swords: “Hell is not empty, vow is not Buddha. The world we live in is notwithout danger and profit. But the reason why there is no collapse can alsosupport many ordinary people to live peacefully is because there areintelligent people who are willing to make sacrifices to melt the world’s iceafter realizing the world’s cold and despair.”ConclusionIt seems that 3 and 4 cannot be eliminated in a wide range of conditions, whichseems to have the greatest probability of victory. However, there are only somany conditions that will not happen as long as position 1 is unwilling.However, position 1 will be eliminated in any case. However, he can choose tolose together or give others the chance to win. Position 2 also needs to seethe meaning of position 1. Position 5 also needs to see whether others giveopportunities In general, he has no chance. Based on their “try not to beeliminated and try to eliminate others as much as possible”, they willchoose 20 pieces and be eliminated collectively.AnnexCode（only for reference）://Computer code of C++:#include #include #define DEBUG 1#define choice_push(a,b,c,d,e) choice[0]=a;choice[1]=b;choice[2]=c;choice[3]=d;choice[4]=e#define N 50#define PC 5int sum(int choice[]){ return choice[0]+choice[1]+choice[2]+choice[3]+choice[4];}int print_choice(int choice[]){ printf(
orashshi
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https://orash-shi.github.io/orashacadamy.github.io/2020/07/06/THE-ANALYSIS-OF-THE-GAME-OF-DIVIDING-MUNGS/
The summary of the game of dividing mungsFive prisoners, in the order of 1-5, respectively, grab mung in a sack containing 100 mungs. Each prisoner is required to grab at least one mung. The person who catches the most and the least will be executed, and the person who catches the same number will also be executed. Moreover, they cannot communicate with each other, but when catching, they can touch the remaining number of mungs.Background They are all very smart people. Their principle is toask for life first. 100 do not have to be divided.My Analysis*My perspective is based on the fact that everyone knowsabout others’ level of intelligence. Each person takes a plan, so that when the rest of the people take the best plan,their own survival probability is the highest. If there are multiple schemes to maximize their survival probability and they arethe same, the scheme to kill the most people will be adopted.If my hypothesis is obtainable andright in some ways, then it may lead to a very horrible ending.We defined m_n as the number of mungs the nth people take, and M_n as the sum of mungs taken by the top n people.Lemma 1When n people(1 ≤ n ≤ 3) has taken mungs, if the number of mungs taken by them make 20n < M_n < 95 + n, then the (n+1)th people should take mn+1=min(96 - Mn + n , [(M - 1) / n]).Prove of Lemma 1In this scheme, you can ensure that you will not die. At the same time, people who have not taken beans are the most likely to die. 96 - M_n + n is to guarantee that the last left people can have at least one mung to take, and the one who takes the mungs before the last one will take at least 2 mungs. [(M - 1) / n] is to ensure that you get at least 1 less mungthan the person who gets the most.Due to the fact that Mn > 20n, we have [(M - 1) / n] ≥ 20. This not only ensures that the number of beans you take is not the most, but also that no one else can get so many beans, so you can survive. If [(M - 1) / n] < 96 - Mn - n, he kept the number of beans left to a minimum while ensuring his own survival, so that more people could be killed. If [(M - 1) / n] < 96 - Mn - n, he can only take one bean for each of the remaining people to ensure his own survival and kill all the remaining people.Corollary 1If the first person wants to survive,he can’t take more than 20 beans. Otherwise, if the latter person takes lemma1, he will ensure his own survival. At this time, the first person will die because he takes the greenest beans, and the last 1-3 people (according to the number of green beans taken by the first person) will die because he takes the least number of beans.Lemma 2When n=2 or 3, if Mn ≤ 20n, then the (n+1)th people will take mn+1 = [Mn / n + 0.5] mungs to ensure his survival possibility.([Mn / n + 0.5] is the rounding of the mean)Because if and only if in thiscase, as long as the difference between the maximum and minimum number of mungbeans taken by the front person is not less than 2, they can ensure their ownsurvival. (Otherwise, the survival range will be narrowed)For the fifth person, this condition may not be true. For example, seeing that the first four people took 62, maybe 14+16+16, or 15+15+15+17 ,so whether he took 15 or 16, he had a chance but could not ensure his survival.And the difference between the maximum and minimum value of mungs taken by all people is not greater than 1, and all people have to die.Lemma 3When everyone is extremely selfish, the first two people are not likely to survive.This is because by lemma 2, if the third to fifth people will take the plan of survival probability for them, if the difference between the number of mungs taken by the second and the first people is more than one, then the first two people have the maximum and minimumvalue, and must die, if the difference is not more than one, then all people must die.ResultsSince the first person has no survival probability, his goal is intriguing:If you don’t have a chance to survive—Option 1: kill as manypeople as possibleOption 2: save as manypeople as possibleAccording to my hypothesis, it should be the former. Since the first person doesn’t havea chance of survival, let’s all die - Take 96 mungs!But if the first person is a little compassionate and choose 2. Then, he will take 21-33 beans. According to lemma1, the 2nd-4th people will survive.Therefore, according to the different understanding of the question, there are two solutions: 1. All people will die. 2. The 2nd to 4th people will survive.For the first prisoner, he will face a philosophical dilemma:If you can’t live, will you chooseto let others bury you or let others live well?The further analysis is in the part “Moral Analysis”.Theoretical AnalysisI have to point out: because there is no “everyone knows that other people are also very smart” condition in the title, there will not be 96 beans selected by A.Let’s see a very basic question that there are only 3 people ABC, 10 mungs, and all other condition is the same.At first, B is very nervous, he began to think.For him, there are 3 tactics:Better : He himself alive.Medium : All the people die.Worse : He is dead, but someone else is alive.Then he began to predict A’s behavior:If A takes 8 beans, B takes 1, C takes 1. All dead.If A takes seven beans, now it’s B’s turn. If B takes 1 bean, C dares not take 1, and must take 2. C survives and the rest die. If B takes two beans, C can only take one. B lives alone. Because B is a rational person, B will take two. A and C are dead.If A takes six, B takes three; if A takes five, B takes four. It’s all B’s and both A and C are dead.B has found the rule, that is, let 9 the number of his own take between AC, he can guarantee life.If A takes four, now it’s B’s turn. If B takes five, C can only take one. A live alone and BC dies. If B takes four, C will die with three people no matter how many beans he takes. If B takes three, C chooses to take three when he knows that the first two peopletake seven, and the three people die together. If B takes two or one, C will choose to take three, and C will live alone. A and B died. B surprised to find that no matter how he chooses, he will die. He won’t choose to let C live. So, B chose to take three.If A takes three, B thinks a little, and chooses three people to die together.If A takes two, B will take three, but C laughs. He doesn’t take five, nor four, nor one. He takes three. All three died together.If A takes one, B will choose one, all three dead. If B chooses 2, C will choose three dead. If B selects 3 or more, and C select the average number of A and B. A and C will live, but B will die, so B won’t make this choice.But A also thought about the wholeprocess. A discovered a sad fact. If B is very smart, no matter how he chooses,he will die. In this case, A places his hope on that B is not very clever. Hesmiles and chooses four beans. Then A, B, C all dead.Then We conclude a theorem: If three people have n beans, n > 3, and A doesn’t know whether B and C are rational, he can choose [n / 3]. If all three ABC people are rational, they will die together. Then if the D and F participate in the game, I myself wrote a code in C++ to analyze all the possibility. It is the same.(See Annex)Let’s see the result:If all of them are rational: 100 left for AA will chose 1090 left for BB will choose 1179 left for CC will choose 1168 left for DD will choose 1058 left for EE will choose 10 This based on that they are abusolutely clever.And obviously, they all dead. Moral AnalysisWe will find that among all the possibilities A’s life depends on B.In other words, if A is smart enough, he will think that his life and death may be decided by B. For example: A grabs 5, B has a way to make A live (grabs 90).But, this kind of decision, needs a premise, name: B has the gratitude mentality.Let’s define the gratitude mentality: Weak gratitude mentality: If others show good tome, on the premise of not affecting self-interest, I choose to be good to him. Strong gratitude mentality: If other people choosenot to be bad to me when they can be bad to me, on the premise of not affecting self-interest, if I can be bad to him or not, then I choose not to be bad to him.As the gratitude mentality is real in the world, A has a great chance of survival.As long as A doesn’t kill B, andlets B go. Although B knows that he will die, A will live as long as B has astrong gratitude mentality.But in the world, the weak gratitude mentality is relatively common, and the strong gratitude mentality is relatively less. ——If I live, let me be good to you. If I dead, I don’t care if I’m good to you.If A is a mediocre but not despicable person (only seeking self-interest, not seeking harm), then his successor B will enjoy the dividend of the forerunner when he has a strong gratitude mentality. Otherwise, A will be the first to die.In other words, A’s survivaldepends on B’s strong gratitude mentality. Whether BCDE must die or not dependson whether A pursues harm.Therefore, in the real social model(under the assumption of self-interest but not necessarily at the expense ofothers), A will not choose to catch 96 and let everyone die.However, no matter what happens, Bwill die. It’s inevitableSo, let’s suppose that A is an elevated person. Also, let’s define the nobility first.Weak noble:If you can be self-interest, self-interest. If you can’t be self-interest, it’s better to benefit others.(this definition is not strict, because sometimes self-interest involvesharming others. The strict definition is too complex, so it is omitted. Inaddition, gratitude mentality is also a specific example of weak and noble.)Strong noble:There is no difference betweenbenefiting people and self-interest.Strong and noble rarely exist inthe world, generally only in close relatives or people with religious beliefs. Weak high is relatively common.Assuming that a is a weak and nobleman, he realized that he could not escape death without other noble people inthe world. Since both sides are dead, it’s better to be a noble person.So, A choose to catch 1. This means thata has the least chance of sacrificing others by himself. If B is vulgar, he will choose two. C, Dand E are all vulgar people, only two of them should be arrested. (1, 2, 2, 2,2) all dead. A willing to save the world, but failed. But as long as there is one person in B, C, D and E who is willing to catch 50, he can save all peopleexcept himself and A. A’s death is to be benevolent. His own death is tosacrifice his life for righteousness. Because two people choose to sacrifice,others can be saved.At last, I’d like to quote Ksitigarbha Bodhisattva’swords: “Hell is not empty, vow is not Buddha. The world we live in is notwithout danger and profit. But the reason why there is no collapse can alsosupport many ordinary people to live peacefully is because there areintelligent people who are willing to make sacrifices to melt the world’s iceafter realizing the world’s cold and despair.”ConclusionIt seems that 3 and 4 cannot be eliminated in a wide range of conditions, whichseems to have the greatest probability of victory. However, there are only somany conditions that will not happen as long as position 1 is unwilling.However, position 1 will be eliminated in any case. However, he can choose tolose together or give others the chance to win. Position 2 also needs to seethe meaning of position 1. Position 5 also needs to see whether others giveopportunities In general, he has no chance. Based on their “try not to beeliminated and try to eliminate others as much as possible”, they willchoose 20 pieces and be eliminated collectively.AnnexCode（only for reference）://Computer code of C++:#include#include #define DEBUG 1#define choice_push(a,b,c,d,e) choice[0]=a;choice[1]=b;choice[2]=c;choice[3]=d;choice[4]=e#define N 50#define PC 5int sum(int choice[]){ return choice[0]+choice[1]+choice[2]+choice[3]+choice[4];}int print_choice(int choice[]){ printf(