For those of you not familiar with what has been called “the hardest logic puzzle in the world”A, The Blue-eyed Monk problem or Blue-eyed Islander problem (Version 2), check out the above reference links. For what it’s worth, that “hardest” title is probably an exaggeration. But my new version, inspired by recent events, might be harder.
In this version of the famous logic problem, there are a number of monks on this remote island. All of them are “perfect logicians”. Every monk sees every other monk every day, but they are rugged individualists who never talk to each other or communicate in any way. But they also take full responsibility for their actions and obey all rules and protocols (yes, this problem is entirely fictional). Strangely, there are no mirrors or reflective surfaces on the island. The ferry visits the island every night to drop off supplies and would take the monks to the mainland, if necessary. But none of the islanders wants to leave and no strangers are allowed on the island. In this version of the problem, all of the monks have red eyes.
One day, almost all of the monks noticed that one of their peers had blue eyes, but they all continue to go about their lives. Many days later, it is noticed (by almost all of them) that two of the monks now have blue eyes. Still, life goes on. This trend continues with a slowly increasing frequency until many, many days later, the first monk to have developed blue eyes is found dead. Doctors on the mainland soon discover that some as-yet-unknown, but contagious pathogen hit the island and the number of monks infected has been growing 2% every day for quite some time. The only visible symptom is the changing of eye color from red to blue. (No other eye color is affected.) The disease is not contagious until the eyes go blue but then remains contagious until the infected person dies. How it spreads is still unknown, but obviously, direct contact or even close proximity is not required. By the time the doctors figure all of this out, red-eyed monks have started changing eye color at the rate of one every other day and the second blue-eyed monk has just died. If a blue-eyed monk gets to the mainland in time, they can be cured, but the ferry does have a limited capacity. The monks are pacifists, so there will be no shooting of blue-eyed monks (yes, I might have a personal stake in this directive). All of this information was left on a large sign at the ferry dock the next day.
So what do the monks do? Will they go extinct? Will 2020 see the end of all logical thought on this planet? Only you can answer that question. Good luck! Leave your answers in the comment section below.
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In the interest of full disclosure and transparency, I have addressed this problem before. After much fumbling, I did come up with a better (faster) solution, which was widely regarded as cheating (see comment to “The Blue-spotted Monks Revisited”.
Above is one version of a common billboard along the interstates and possibly other highways in Florida and across America. The problem for me is that the spokesman didn’t look anything like that when he was 18 days from conception; he would have been barely visible and a lot less cuddly. Below, I’ve taken the liberty to replace him with his earlier self, which would have been about half the size of a pea or just able to fit through the pupil in the eye of his older self as shown in the background of the revised billboard. His heart would have been larger than a grain of sand. He still has no eyes, arms, or lungs, and I’ve found no information on brain function. I don’t think he’s feeling any pain. Now ain’t that cute?
His Odds Of Success
While I have touched on this subject before (in my article Save Your Birthday – Vote For Clinton), this time I dug deeper in order to put things into a more serious perspective. Here’s what I found:
According to ABORT73.COM, abortions have been decreasing slightly every year for the last decade, with 879 thousand occurring in the United States in 2017.
The Statistica website says that the number of births in this country has been fairly stable. There were about 3.86 million of them in 2017.
Here’s what that means: if you start with 18 fertilized human eggs, 13 will die from natural causes before birth, and one embryo or fetus would be aborted. Both forms of attrition are “front-loaded”; 2/3 of abortions happen by the 8th weekA1, while 2/3 of fertilized eggs will die in their first 11 weeksA2.
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A common mistake would be to compare the number of abortions with the number of births directly but to compare apples to apples, as they say, you need to use the same reference point. Here’s the math:
3.86 million births divided by 25% (or 0.25) gives 15.4 million fertilized eggs. That would be the common reference point. Divide that by 879 thousand abortions and you get the total number of fertilized eggs there are for every abortion (17.6 – let’s round to 18). To find the number of those fertilized eggs that would die naturally, simply multiply the same 18 by 75% (or 0.75).
One thing that is not clear is “what are the odds that the aborted “child” would have died from natural causes anyway?” The easy answer (and my best guess so far) is 75%. Which means there are a lot of self-righteous people out there (many being middle-aged white males with nothing invested) who are willing to throw a woman in jail for long periods of time without even caring about the circumstances of her case to protect something that probably wasn’t going to make it anyway. I guess for the pious, any excuse is a good one for beating up on your neighbor.
As a teacher, I didn’t heartily endorse “the Curve” because I didn’t believe that just because one of my students could find five friends who were either stupider or lazier than they were they deserved a free pass (although I’m sure that for some of my students that was no trivial feat). This also tied into a theory I developed early in my teaching career, a prophesy perhaps, that unlike doctors, who are said to be able to bury their mistakes, if teachers were to make mistakes, and for political or social reasons were unable to correct the system to reduce the number of those mistakes, non-educated people could eventually reach critical mass, and could then elect enough unqualified politicians to negatively affect the education process (with budget cuts, for example), which would be the beginning of a feedback loop that could put American education and thus American society on a death spiral. But that could never happen here, right?
Now that Mr. Trump has been elected president and his conduct, unchecked by congress, has become more and more outrageous, I find it interesting to watch some of his supporters as they have to warp their world (or the principles by which that world is defined) to greater and greater degrees in order to be able to justify His actions within that small world. To do that successfully, they need other people (their five friends) to be dumb enough to fall for those contorted explanations. As time goes on, those friends need to recruit their own set of even dumber friends so that they don’t look like the fools (that they are) . . . and so on. But even now, dumb and foolish people are not an unlimited resource in this country, and Trump’s actions will continue to test that resource like no other leader in the history of the Earth. So as in all such schemes, the buffoon bubble is bound to break. The only question is – will that happen before the next election? I’m not so sure it will. Any comments?
A long time ago (when I was in the sixth or seventh grade) in a galaxy far, far away (namely Southern California), I was introduced to Einstein’s Special Theory of Relativity by way of a story about two astronauts on different spacecraft watching a bouncing-light-beam clock, and I was really impressed. But as I was chewing my cud (See definitions of “ruminate”) as I was wont to do, and walked home from school, it didn’t seem to make as much sense upon further review. I developed some questions but didn’t know where to get answers, and as life pressed on my attention wandered elsewhere, and everyone lived happily ever after. . .
Until recently. In the last year, the subject has come up several times, the questions seem to be the same, and I still don’t know where to turn.
Although not exactly as I remember it, www.dummies.com1 describes a similar thought experiment in the second section, “UNIFYING SPACE AND TIME”, with a spacecraft traveling at ½ the speed of light, but doesn’t give much explanation. A more detailed explanation can be found in The Star Garden2.
“Time Dilation”, Section 7.2.2 of Reference 2 concludes
“The time between heartbeats is also slower, and so from the perspective of a stationary person, a moving person appears to be living their life at a slower rate. Conversely, from the perspective of the moving person, the stationary person seems to live their life as if it is being fast-forwarded. If they travel fast enough, then they could see the stationary person age before their eyes.”
My problem with that conclusion is that based on the second paragraph of Section 7.1 at the beginning the article, which states
“there’s no such thing as absolute speed or velocity, and something can only be said to be moving at a constant velocity relative to something else. In the same way, something can only be said to be stationary relative to something else”,
how do we really know which astronaut is supposed to be aging before our eyes? What if we put a bouncing-light clock on each spacecraft? Would it explode?
The authors of Reference 2 seem to address this issue at the bottom of the next section, 7.2.3, where they say
“The twin paradox
The twin paradox asks why the astronaut can consider themselves to be moving and the Earth to be stationary, when Galileo’s relativity shows that there’s no such thing as absolute velocity. Why can’t the astronaut consider themselves to be stationary while the Earth moves away at tremendous speeds?
The answer is acceleration. Galileo’s relativity applies to inertial – that is non-accelerating – reference frames. The fact that the astronaut must have accelerated before getting to such high speed means that they know they are the one that is moving.”
To me, this sounds bogus; any acceleration before or after the experiment should be immaterial. Let’s have three observers; one person remains on Earth while two astronauts board different spacecraft, each leaving the Earth in opposite directions and reaching similar stable speeds well in excess of ½ the speed of light (meaning their relative speed would exceed the speed of light in a non-relativistic world). Each of the observers has their own bouncing-light clock. If you start counting after their speed stabilizes, exactly how do each of the three observers see the ages change for the other two?
One Last Question
A question that one might ask in each of these scenarios is “how does the state of the bouncing light in one spacecraft become known to the other observers?” Reference 1 states that Amber, on a different spacecraft, would see the bouncing light travel further between bounces, as if Amber had super X-ray vision and/or was otherwise experiencing the light beam in real time. How does that work? If she had to wait for reflected light rays from the event to reach her eyes, would that affect the apparent outcome in any way?
One phenomenon that may or may not have anything to do with the solution to this problem involves ocean waves. In deep water, a wave’s speed is nearly proportional to the square root of its wavelengthA,
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where S is the wave’s speed (measured in meters per second) and W is its wavelength (in meters).
For shallow water waves, the speed is proportional to the square root of the depth.
where d is the water’s depth (in meters).
but in all cases, it is much less than the speed of light. If an observer were to watch the crest of a wave as it moved along a seawall, or along any imaginary line that wasn’t along the wave’s direction of travel (directly away from a point source, or in the direction of the wind, or perpendicular to the wavefront, etc.), then the apparent speed would be greater than the calculated or expected speed, and as the angle of that reference line approached 90° to the direction of travel, the apparent speed would approach infinity.
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where A is the apparent speed, S is the expected speed, and θ is the angle between the reference line and the direction of travel.
which is well above the speed of light. As far as I know, this has no implications or gives no reason for hope for wannabe time travelers.
So now you see my dilemma. To repeat the title plea, please help me understand. A crucial early step in solving any problem may be asking the right questions. Finding those should be as important, and in some cases may be as difficult as answering them. So let’s get started. Thank you for your help. If I do figure it out without your help, I’ll let you know.
Many companies have done matching-donation campaigns, whereby if you give a dollar, they will give a dollar. It can be effective in promoting more charity by the public in some communities. It too can be a gimmick. It could just be an excuse to limit their donation (we can afford to budget one million dollars, but if you can’t find matchers, I’ll just pocket the difference). Usually they mention a cap or maximum donation that they are willing to make. That’s where the real deceit comes in.
These limits sound reasonable for making sure the donor company doesn’t get in over their heads by misjudging popular support and become committed to a donation they cannot afford, but these companies have already done their homework. For example, if the Green Cross (names in these examples have been changed just to keep you from thinking these principles have limited applicability) Christmas campaign raised ten million dollars last year and has grown as much as 20% year to year, with many years much less, an observer would have reason to be cynical if the donation-matching company has a donation limit of much less than about 15 million dollars. Typically, the donor company will impose a limit of one or two million dollars. I’m sure their maximum donation will be much less than the cost their advertising company would charge for similar exposure. But when the donation-matching company imposes such a low limit, their actual final donation will always be equal to that limit, reaching the limit will NOT be advertised, and the deception of the public will begin. The ethical thing would have to just announce they were making a simple donation of one million dollars, but if you can’t fool suckers into coughing up more money (mistakenly thinking their donation is worth twice as much to the charity), where is the sport in that. At least this time it is for a good cause.
Walmart Doubles Down
Walmart’s new wrinkle takes this scam to the next level. But all it means is that the owners of Walmart will reach their company donation limit sooner, and will be able to laugh at the stupidity of even more Americans. That’s called arrogance.
Suppose that in this year’s campaign, the Green Cross is expecting ten million Americans to each donate a dollar (I’m just making the math simple). Under their old plan, Acme Widgets would make their generous four-million-dollar-donation by announcing they will match your donation dollar-for-dollar (for up to four million of their dollars). The first four million Americans bring in eight million dollars for the Green Cross (including Acme’s portion), but if the remaining six million Americans donate expecting to see their money go further, they are being misled. They will bring in just six million dollars, and the Green Cross total for that year will be $14 million – a clear victory for the Green Cross (and Acme).
Under their new plan, Acme will now make their budgeted four-million-dollar-donation by announcing they will match your donation with, say, four dollars for each of yours (for up to one million of your dollars). Now Acme reaches their limit much faster, as the first million Americans, with matching funds, help the Green Cross bring in five million dollars, while the last nine million donors (up 50% from last year) will make a contribution that they mistakenly believe is being amplified. They will contribute only their own nine million dollars, bringing the Green Cross the same $14 million as last year, including the same amount from Acme and the same amount from the individual donors, 90% of whom were deceived.
I am not suggesting here that donating to charity is bad; quite the contrary. I believe helping others is a good idea, an idea that is even supported in the Bible. But there are a few things one should consider:
Are you donating for the right reasons? Donations with strings attached aren’t really donations. You should not donate expecting to get anything in return (not even a reserved seat in the Hereafter). Your rewards should be internal.
Similarly, don’t assume, think, or claim the recipient of your donation will place a higher value on it than you do; that’s just fraud. I’ve recently heard complaints from members of relief organizations accepting donations for victims of Hurricane Harvey about people donating expired or opened food, worn-out clothes, etc. You are not fooling anybody.
On the other hand, are you giving within your means? Creating a hardship for one person to ease another – the saying for that is “Robbing Peter to pay Paul” (a Biblical reference, but not from the Bible), is not recommended. Even the airlines advise you to put the oxygen mask on yourself before tending to your children.
Finally, not all charities are created equal. And not all charities are the best stewards of your money. I suggest doing your homework;
After checking out a charity’s website, you can research potential donees at places like Charity Navigator.
As time permits, I will be discussing some ways advertisers use math to deceive their customers. Stay tuned.
In the works, I have two different questions for you:
‘Do Medium-sized Egos Really Exist?’, and
‘Should Law Enforcement Officers Be Allowed To Use The “I was afraid for my life” Defense?’
Both of these require some preparation/research, but I hope to have them ready before too long. For now, I’ve chosen a lighter topic about a scheme that, because it’s not being implemented as designed, could well be simplified.
When I started school, we got one grade, from A to F (I never learned why E was left out), to represent our mastery of the subject. Then, at some point, they introduced a separate grade for effort (from 1 to 3) and another for conduct (also A through F without the E); these were promoted as independent variables that could give more insight into the performance of one’s child. I soon had reason to question the independence of these variables.
What’s The Best Grade You Can Get
Conventional wisdom tells us that the highest grade one can get would now be an A1A. I’m not here to discuss the merits of bad behavior, so we will focus only on the first two symbols. To me, it was obvious that an A3 would be more desirable. Here’s why –
Suppose it’s a leap year and you are betting on track events at the Summer Olympics. In the first heat, the first place runner comes in with a time of, say, 4:00.00, and at the end is visibly spent (lying on the ground, breathing heavily, and sweating profusely). Her grade would clearly be an A1. In the final heat, the winner has the exact same time but isn’t even breathing hard. I would give her an A3 (keep in mind that it is not uncommon for runners at big events to pace themselves – save some effort if they can afford to, for later heats). Of course, both runners advance to the finals. Again, conventional wisdom gives the higher grade to the first runner but tell the truth – which one are you betting your hard-earned money on in the finals?
So you can see what grade I was trying for. But the truth is teachers don’t give A3 grades, even if you never turn in your homework. This isn’t a case of political correctness (whereby we fashion our remarks based on the possible objections of imaginary people with hyper thin skins or real fools priding themselves on how easily offended they can be), but another common problem in the political arena whereby people refuse to let facts get in the way of their idea of the way things should work in their perfect (but grossly oversimplified) world. In their view, the very fact that you got an A proves that you were trying really hard because hard honest work is what made America great. The problem is once you make that link between those previously independent variables (effort and results), then you are really only working in a one-dimensional world and don’t need two grades to adequately describe it.
Looking From The Other Side
But you may be saying to yourself “Silent, you are the anomaly! Only the very rare person who can find a task at which they can succeed without unbelievable effort would have the luxury of taking your position on this topic”. If you really think failure is the norm, then answer this. Do you really think someone who, for whatever reason, didn’t meet the minimum requirements for success in this class, would prefer an F1 over an F3? From what I’ve observed, the opposite has usually been the case. If you give him an F1 you are saying “bless his little heart, he gave it his best shot but is just too stupid to make the grade”. Giving him an F3 gives him an alibi (or more accurately, reinforces the excuses he’s been giving even without your blessing) that he’s really very, very intelligent, but just didn’t put forth the effort.
There are two ways to cure this problem: we could start treating effort and results as the independent variable they are (which is probably too agonizing a task for most teachers) or just stop giving the effort grade. I propose the latter. What do you think?
Earlier this month, the Catholic archdiocese of Newark, New Jersey, decreed that after four years of Catholic Youth Organization (CYO) basketball together, the St. John’s 5th grade team (nine boys and two girls) would not be allowed to play the last two games of the season with girls on the teamA.
First, A Little Math
The maximum of any subset cannot be greater than the set maximum. This means that if the largest member of your weight-watching group is, say 400 pounds, then as people leave the group, that maximum will not get instantaneously larger; it could remain 400 pounds for a while, but will probably eventually get smaller.
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The average group weight, on the other hand, could go up or down depending on how people are selected for removal from the group, but averages aren’t generally used to measure greatness.
Similarly, in sports you cannot raise the level of competition by restricting participation, meaning that you can’t say your team is the best in the universe if any member of the universe was barred from competing. Consequently, the only logical reason for restricting membership would be to protect those members from unfair competition, meaning a team would only ban girls if they thought their boys weren’t ready for real competition. As we all know, a group’s stated reasons for an action may differ from their real reasons.
I suspect the archdiocese’s advertised reason for the decree is to protect girls from competition they can no longer handle. But for that argument to have any credibility at all, at least two new conditions would have to be in effect:
There would actually have to be a girl’s team if you want anybody to believe that their interests are really your first priority.
You would protect a “weaker” group by banning the unfair competition from that group, not banning the allegedly weaker competition from the “stronger” or open group. The later option will rightly cause others to question your motives. “Who are you really protecting?”
The required game forfeitures would be further evidence of their true motive. A team is required to forfeit a game only if they won using an unfair advantage. You would not make a boxer forfeit all the matches he won with one hand tied behind his back. Obviously, the other boy’s teams not only considered the girls a threat, but most likely the sole reason for the team’s success.
A Happy Ending
On hearing the decree, the St. John’s 5th grade team immediately and unanimously decided to stick with their teammates and forfeit the season.
The girls, understandably, felt bad and offered to sacrifice themselvesA. St. John’s athletic director honorably rejected that offer (although in the body of that article, it suggests that the league director had already cancelled St. John’s season, making the athletic director’s gesture moot.)
A new Cardinal reversed the ban and allows St. John’s to playA.
A Not-So-Happy Ending: Politics Trumps Logic
I just read about a different, but logically related case in TexasA, where a girl was taking testosterone to become a boy and wanted to compete with boys, but was required to compete as a girl and won their state wrestling championship. I’ll leave the application of principle and subsequent comments to the reader.
You are probably very proud of your grasp of English (unless you live in South Florida, in which case you may not give a damn). And yet I have seen plenty of people whose lack of understanding about basic structures like the “If” statement
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The next article in this series will discuss how misunderstandings about the conjunction “or” have caused so much trouble.
cause them to make terrible assumptions.
Suppose a young child is misbehaving to the point that the caregiving parent decrees “If you don’t knock that off, I’m going to paddle you” (This is an old example; I’m sure nobody would ever actually do that today 😉 ). As young children have been known to do, for whatever reason, the child continues with its behavior. The parent repeats the statement, with added emphasis. Nothing changes. The parent soon throws their hands up and says “wait until (your other parent) gets home”.
The parent’s first decree, like all “if” statements, had two parts; a condition and a consequence (joined by the conjunction “if”), with the understanding that if the condition is true, then the consequence will occur. It’s simple enough that even a young child can understand it. If the condition is met and the consequence is not accomplished, then the statement would be considered false. In short, the child knew that the parent was lying.
Now suppose the non-caregiving parent comes home, sees the objectionable behavior, makes a similar decree, and then the first parent points out that they had already made that decree to no avail. The child, for whatever reason, stops the objectionable behavior. To everybody’s surprise, the second parent paddles the child. Although the child and many of you listeners may think bad thoughts about this parent, one thing you can’t call him/her is a liar.
As you can see here, the problem with the “if” statement is that is incomplete in the sense that it only addresses what happens when the condition is true, remaining completely silent to the possibility that the condition could be false. This allows most people to make the assumption that if the condition is false, the opposite of the consequence must occur. As the young child in our example learned, that assumption would be a mistake.
Don’t make stupid assumptions. As your lawyer would tell you, get it in writing. In the above example, since the second parent didn’t make any promises about what would happen if the behavior did stop, s/he can’t be accused of lying. If this example bothers you, I’m sure the second parent told the child afterward that the paddling was for not obeying the first parent, in which case we would be unable to judge the truthfulness of their claim until after the right set of conditions are met following some later episode of misbehavior (guesstimating any change in likelihood of that future misbehavior based on recent events will be left as an exercise for the reader). To lawyers, mathematicians, and the like, the parent’s explanation doesn’t matter to this case and is unnecessary.
Logicians have named operators (or functions) fulfilling all sixteen patterns of truthfulness or falsehood of expressions based on the truthfulness or falsity of two variables, such as the condition and consequence of the “if” statement described above. Engineers call the statement that yields the results you thought the “if” statement provided the “exclusive nor” function, “nor” being short for not or, meaning “giving the opposite results than the ‘or’ function”. Some refer to it as logical equality. In English, it would be represented by a sentence including the phrase “If, and only if”, such as “I will ground you for the rest of your life if, and only if, you do not stop screaming this very second”. If that type of statement had been the norm in this household, the non-caregiving parent, upon hearing the lack of results achieved by the other parent, was still free to add other (most likely “or”) clauses (to be discussed in a later article) to his/her decree. If you are now totally confused, please do not sign any document containing more than six words before consulting an attorney, or at least a mathematician. On second thought, in cases like this, I would stick with the lawyer.
If you are not aware of the Blue-spotted Monk problem, or haven’t seen my previous post (in which I made an error in logic), please visit The Problem Of Blue-spotted Monks.
Did you ever wonder what would happen if, as a cruel joke, it had been reported to the monks that one of them had the dreaded blue spot disease when in fact nobody was afflicted? (Am I really the only one here with a slight sinister streak?) Since none of the monks saw a spot elsewhere, they would each assume it was them that was infected and the very first day all the monks would have started to gather at the exit station. One of them, as he approached, would notice that none of the others had a blue dot and would start to chuckle to himself, thinking “these clowns can’t even count”. The others, hearing him laugh, would look around for the first time (until then they were so sure of their logical talents that they hadn’t even bothered to check), and by the time he stopped laughing the rest would have disappeared into the forest. Then it would occur to him that maybe he didn’t have a spot either. When word got back to the Guru, the laugher would be expelled for violating the “no communication” rule, but it wouldn’t affect the solution because the other monks only counted the peers that had spots on their forehead and he was never missed.
Plan B (I named this scheme in honor of B, who first put things in perspective for me. If this plan turns out to be flawed, like the last, I take full responsibility.)
When answering nature’s call in the middle of the night a few days after accepting B’s explanation about why the monks had to start at zero, it struck me that since a monk was really only concerned with three possibilities in the number of blue spots anybody has seen, modulo arithmetic might give a way to synchronize everybody’s universe or get everybody on the same page, so to speak. Plan B calls for everyone to start counting not at zero, but at the last multiple of six. For example, the person who saw ten blue dots knows that there are either:
ten afflicted, not including himself, which means there are ten people who see only nine spots. Everybody else sees ten.
On the other hand, if he is blue-spotted, there are ten others like him who see ten spots and everyone else sees eleven.
Those who see nine spots know that those who see eleven don’t really exist, and those that see eleven know that the nine-seeers don’t exist. The monk that sees ten must consider each of the other two cases, but not both at once.
Under Plan B he would start counting at six, as would the possible people who saw nine, and those potential people who saw eleven. If everyone knew to start at six, the nonexistent people who saw only six spots would be gone before the second day (we’ll discuss them again shortly), and since that won’t happen, the nonexistent people seeing seven spots will leave on Day 2, etc. Our guy knows he can sleep in until Day 4, when the really possible nine-spot sighters would be scheduled to leave. If he saw spots on Day 5, he would turn himself in and everybody else would live happily ever after.
Starting the inductive thinking process, the first six possibilities start counting at zero, just like the old days. We’ll call that their landmark. As the logic countdown continues past the next landmark (which would be six in this scheme), the hard thing for me was knowing who would start using it first, or even if it was possible to make the switch if that person needed to wait for the person ahead of him (who is still using the old landmark) to make his move. I had the hardest time reconciling the notion that you needed to wait for the people ahead of you with the notion that those people don’t exist. It turns out the solution was easier than I imagined.
In Plan B, if the number of spots you see happens to be six (or any exact multiple thereof), your dilemma is that the possible person who sees five spots started counting at zero, and won’t budge until Day 6. The guy who sees seven spots will start counting at six, and is depending on you to leave now (Day 1) or else he’s lost. Go to the exit point on Day 1. If you are not infected, there will be a very large crowd heading for the exit with no spots on their head while six people with a spot sit comfortably at home. When you spot your peers, you will avoid the crowd and head home knowing that those six afflicted people will all leave on Day 6 as scheduled, and all is right with the world. If you do have the spot, there will be six others with spots heading out that day with you while everyone else waits patiently at home. When those who originally saw seven spots wake the next morning, you will be gone and the problem will be solved.
Lately this problem has turned into an on-again-off-again obsession for me. I first saw a different version of the problem when I was much, much younger; I didn’t figure it out for myself but the answer made perfect sense. Just a few weeks ago, I ran across the spotted monk version I discussed in my last post and again, when reading the answer, I was completely satisfied (although I felt for the commenters who couldn’t accept that it would take 100 days for 100 blue-spotted monks to turn themselves in). Days later, while on a walk contemplating even more difficult (but unrelated) relationship issues, an answer just came to me out of the blue, which is the answer I last posted. It was flawed, and when B. gave his “parallel universe” explanation for why we needed to start at zero, I thought it made even more sense than the conventional answer did, and I was again happy. Several days after that, the modular arithmetic idea just came to me (as discussed). Then it was only my day job and other commitments that slowed me from working toward this solution. But we are not actually finished yet.
Calling All Logicians
All I’ve done so far (if I got it right this time) is to show that starting at zero is not strictly necessary. Plan B is not the only possible plan, however. In fact, I doubt it is the best or fastest plan (I suspect that using a smaller modulus might be helpful). For the monks to abandon zero as their global landmark there would have to be an understanding that there was a single, logically optimal plan to replace it. Based on my difficulties wrapping my mind around the issues so far, I don’t feel I qualify for the “perfect logicians” requirementA of this monastery. It’s just as well; as an aging curmudgeon, that vow of silence probably wouldn’t have worked for me for very much longer anyway. I’m hoping that the real experts will take it from here and God will finally allow me to let go of this problem. If you do see a flaw in this scheme (a check of the references will show it would not be my first mistake on this problem), let me know. I may not be able to fix it, but in keeping with journalistic standards, I am willing to admit and advertise the error. Thank you.
I recently ran across the Problem of the Blue-spotted Monks again at https://richardwiseman.wordpress.com/2011/04/04/answer-to-the-friday-puzzle-98/. Actually, this is a slight variation of the well-known Blue-Eyed Monk problem that can be found at https://en.wikipedia.org/wiki/Common_knowledge_%28logic%29, among other places. One site even called this problem “The Hardest Logic Puzzle in the World”A, but that was probably just a case of self-promotion. For today’s discussion, I chose the first version of the problem because it is simpler (we don’t have to worry about the case in which one person has red eyes). If you haven’t done so, go ahead and read the problem. We will discuss the solution in the next section.
Start With Induction
The classic answer uses mathematical inductionD to first consider the (trivial) case in which only one monk has the disease. Then they move on to the case of two monks. Their error begins in the case of three infected monks, assuming that the monks were required to start back at one in making their individual analysis. That is wrong; in the comments section of some of the references, several people take issue with this assumption. I believe this to be a misapplication of the induction process. The monks have only to consider two possibilities; either they are infected or they are not. For the sake of argument, let’s say there are ten infected monks. Most of the monks will see ten fellow monks with blue spots. Ten of the monks will see nine monks with blue spots. None of the monks will see only one monk with blue spots, or just two monks with blue spots, etcetera. Those monks seeing ten spotted monks have only two possibilities – either there are ten infected monks or there are eleven. Those monks seeing nine spots need only consider the possibility that there are either nine or ten monks infected. Most of the monks, in considering their first possibility (that there are only ten infections) realize that those ten, in considering the possibility that there are only nine infections, must allow for the possibility that there are nine monks who see only eight infected monks. With the information available, nobody sees any reason to consider any other lesser possibility. Just like in the conventional solution, each monk, having two possibilities, must allow the lesser possibility (which means they are not infected) to resolve itself first before concluding that they are infected and turn themselves in. All of the monks know that the least number of infections that any of them must consider is eight, not one. The first day nothing would happen. The second day, the monks that saw only nine other spotted monks will conclude that the possibility of anyone sighting eight spotted monks was groundless, so they will all turn themselves in. The third day, those who saw ten spotted monks will all breathe a sigh of relief.
Considering The Second Possibility
Now we need to look at the big picture. This isn’t rocket science. Why hasn’t anybody seen the error in conventional wisdom before today? I, like the monks, must consider the possibility that I am the one infected. If nobody else comes forward with a confirmation of the correct answer soon, I guess I’ll be forced to turn myself in. Please hurry.