The Silent Knight- In a fight to the death with "conventional" wisdom

A Better Solution to the Blue-eyed Monk Problem

The Problem

There are variations to this problem, but we will go by the XKCD version discussed in www.xkcd.com/blue_eyes.html. It goes:

“A group of people with assorted eye colors live on an island. They are all perfect logicians — if a conclusion can be logically deduced, they will do it instantly. No one knows the color of their eyes. Every night at midnight, a ferry stops at the island. Any islanders who have figured out the color of their own eyes then leave the island, and the rest stay. Everyone can see everyone else at all times and keeps a count of the number of people they see with each eye color (excluding themselves), but they cannot otherwise communicate. Everyone on the island knows all the rules in this paragraph.”

“On this island there are 100 blue-eyed people, 100 brown-eyed people, and the Guru (she happens to have green eyes). So any given blue-eyed person can see 100 people with brown eyes and 99 people with blue eyes (and one with green), but that does not tell him his own eye color; as far as he knows the totals could be 101 brown and 99 blue. Or 100 brown, 99 blue, and he could have red eyes.”

“The Guru is allowed to speak once (let’s say at noon), on one day in all their endless years on the island. Standing before the islanders, she says the following: ‘I can see someone who has blue eyes.’”

“Who leaves the island, and on what night?”

The Traditional Solution

The traditional/standard/irrefutable solution is described everywhere, including in xkcd.com/solution.html. Basically, if there were only one blue-eyed monk (BEM), then he wouldn’t see any. He would realize he must be it, and leave that night. If there were two, they would each see only one and would assume the other guy would leave that night. When they saw each other the next day, they would both leave. And the people who were worried that they might be Number 3 would all breathe a sigh of relief. And so on. For the case described above, all 100 would leave on Night 100.

Disclaimer (and Personal History)

There is a reason why I was never invited to Logical Island. When I first read this problem, decades ago, I didn’t even figure it out on my own. I cheated and read the solution. But then I thought, “That’s pretty slick”! When I ran across the problem again, a decade or two later, I remembered the answer and still thought it was pretty cool. But I thought it was too bad that there wasn’t a quicker solution. The third time I ran across it, another decade or two later, I became obsessed with finding that solution. It wasn’t going well, but a few nights later it occurred to me that the answer might involve modular arithmetic^Explained. A test using Mod 6 was effective.

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In the Mod 6 Method, the “proof” proceeds normally up through five BEMs. Those seeing four leave on Night 5. But if there are six BEMs, those seeing five (you know what that means) would await their findings on the morning of Day 5 to decide their fate. Those seeing six BEMs, however, would leave the first night since six is 0 (mod 6). Their first clue as they approached the dock would be the dozens of other monks at the dock instead of the six others they were expecting. Even more important, perhaps, is that none of them had blue eyes. They would all have a good laugh (probably against the rules), and then quietly return to their homes. Those who originally saw five (a.k.a. 5 mod(6)) were sleeping when all of this happened and wouldn’t know anything about it. The problem would take care of itself four days later.

The next logical question would be “Why is six so special?” If all of the monks can’t agree on the same base/modulus without communicating, the scheme fails. When I introduced my idea on a different forum, one commenter quipped (sarcastically), ‘if that scheme is even legal, what’s to stop you from just having everyone who sees an even number of BEMs go to the ferry on Night One?’

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I posted a couple of blog articles on the project here, but they received no comments and have been superseded by this one.

I thought that was way too extreme, but as I was typing “ridicul . . .”, a bell went off in my head. Of course. That’s it! Mod 2. But that commenter wasn’t buying it.

The Fastest Answer

In my plan, everyone who sees an even number of BEMs (remember, zero is an even number) goes to the dock the first night. If when they got to the ferry, everyone else there had blue eyes, they would all board the ferry. Otherwise, having non-blue eyes, they would all quietly return home. If the problem isn’t solved that night, everyone that sees an odd number of BEM’s (everyone else) leaves on the second night.

Let’s say there were 99 BEMs (or 101). They would all see 98 (or 100) BEMs, and they would (as in the case of one BEM) get on the ferry the first night. That would be that. But if there were 100 BEMs, those seeing all 100 would all go to the ferry the first night. They see nothing but non-BEMs and all go home quietly. The next morning, those seeing 99 and hoping that those seeing 98 acted accordingly, would realize that wasn’t the case. They would all leave that second night (as in the two BEM case). End of story.

Advantages

If there are 200 monks between the ages of, say, 19 and 85, with minimal turnover, you can expect three deaths each year. That means there is about a 50% chance that the number of BEMs will mysteriously change while the 100-day clock is ticking. (I’ve heard no discussion about how the traditional method handled such things. But a similar situation is addressed in the next-to-last paragraph [Go]). My new method virtually eliminates that possibility. Even if nobody dies, there is over three months of meticulous counting that can be avoided.

Other Concerns and Considerations

More About the Technique

As we’ve discussed, this method is actually based on the traditional method, but folds the number/time line into the smallest possible segments using modular arithmetic. For a monk that sees 100 BEMs, there are only two possibilities. Either there are 100 BEMs (which all see 99) and he is among the 100 others, or he is one of 101 BEMs. Nobody seeing 100 thinks there might be 102 or 99. If you see 100 and think you might be one of 101, then you don’t care what the non-existent monks that see 102 BEMs think. For those seeing 100 BEMs, this scheme folds the other monks from both possibilities – those seeing 99 and those seeing 101 – into heading to the ferry the same day.

As mentioned, the solution must be so logical as to be adopted unquestionably by all monks without discussion. First, remember that they were able to come up with the original method without discussion (something I couldn’t do). Why would they stop at the first answer they came up with when they can clearly see the disadvantages to that approach. This answer adds another simple mathematical concept for the next logical step. But if you know of another logical solution candidate, even completely unrelated to these techniques, speak right up.

I mentioned that there are variations to this puzzle. This technique would not work in variations that require the monks who discover their eye color to quietly commit suicide in their homes. And it is the ferry part, not my new technique, that would minimize the damage of an oracle mistakenly (or maliciously) claiming he saw a BEM when there were actually none.

Questions Answered

Even if based on the original method, doesn’t the mod 2 math allow two possible answers? Why wouldn’t they have those seeing an odd number of BEMs go first? Because the “let odd go first” plan is not the direct result of simple mod 2 math. And the first two cases (1 and 2 BEM’s) in my technique are exactly the same as the original method.

I’ve heard that the inductive method proves that the original technique is the only one possible. Nonsense! All mathematicians know there are commonly more than one way to solve or prove anything. Just a couple years ago, two teenage girls found a new trigonometric proof to the Pythagorean Theorem^Article.

I’ve heard that these monks are illegally communicating information. No, they are giving the same information at night as they were giving during the day – no more information than in the original approach. In either case, when the lone BEM disappears the first night, he is telling every other monk (that was worried that he might be Number 2) that they can stand down.

But what would happen if some of the monks stuck with the old technique? Well, I guess all monks weren’t perfectly logical. But if there were an odd number of BEMs, for example, and only some of those seeing an even number go to the ferry, as my plan dictates, then they would see other BEMs and leave as required. The next day, those who chose the traditional solution would notice that there were far fewer BEMs. Regardless of whether the new number of remaining BEMs was odd or even, they would realize what happened and leave the next night.

(If they still couldn’t figure it out, they could at least proceed with the traditional plan using the new BEM count at the original starting point. [Return to Advantages]

Would they be allowed to take action without knowing their eye color? Nothing in the rules restricts their access to the ferry dock or prevents them from noticing the eye color of those they meet. I would expect logical monks to be curious and willing to conduct simple experiments that could benefit themselves and the group.

Any questions?

Another Solution For Legislative Voting Systems

Our forefathers, in creating a new nation, were concerned about a democracy’s capability to degenerate into “mob rule” through “the tyranny of the majority.” James Madison, who later became our fourth President, in Federalist #10^A and in debates in 1787, argued that the government should protect “the minority of the opulent against the majority”.

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According to vocabulary.com^D means rich and superior in quality (but contains connotations of pretentious). Our forefathers, like Madison, were considered wealthy.

Yet Lord Acton’s belief that “. . . absolute power corrupts absolutely”^A can be seen in states like Florida with supermajorities controlling their government. In 2023, legislatures with veto-proof majorities in both houses^A control 29 states. Florida’s governor, Ron DeSantis, with no resistance from the legislative branch, has waged a culture war of tyranny against “the woke” – Democrats, women, minorities, and anybody that publicly disagrees with his policies – in his bid for President of the United States.

In my article “Two Political Parties Are Not Enough”, I address some of the problems of our current two-party system (promoting polarization, guarantying an instant majority). I propose a solution (more political parties). But then I also discuss obstacles to that solution. While waiting for changes, I’ve come up with a simpler solution.

My New Solution

I propose adding a requirement to the only existing rule of securing a majority vote (or 60% in the Senate, or . . .) when state and national legislatures vote on bills. It should also be necessary to secure at least 25% of the vote from each minority party. This would fight polarization and encourage collaboration, compromise, and bi- or multi-partisanship to find solutions at least partially acceptable to all constituents.

Is 25% the right number? It can be tweaked. And as the rule is stated, it can be combined with other solutions (whether previously discussed or not). Could it get in the way of passing laws in an already dysfunctional government? Well, yes, but that’s actually the point until we can elect representatives that represent all the people while trying to solve problems instead of just making sure the other team doesn’t score any points. They apparently can’t do that individually, but maybe they can collectively.

By now I shouldn’t have to tell you that we accept opposing view and solutions. Start the discussion in the comment section below. Thank you.

I’m Back – Are You Still Interested?

No, I’m not a red-eyed monk (see our last post, Will 2020 See The Extinction Of The Red-eyed Monks?). And just like Samuel Clemens, any reports of my death were exaggerated. Nor did I run out of material to write about. The truth is that my day job grew increasingly hectic, and just as we started to get that under control, first COVID and then personal issues began to occupy a lion’s share of my time.

Not all of that has been completely cured, but I will try to commit to at least four posts a year until I finish getting my act together, so to speak. My next article could be a new idea for how congress passes laws. Also look for an article about what our forefathers meant when they said we were created equal, maybe some comments about our Supreme Court and possibly other constitutional issues, and probably an explanation of math concepts used/exploited in places like advertising. To review our previous articles, see https://thesilentknight.info/. There should be some formatting changes ahead, starting most likely with my code for reference links.

I’ve had to upgrade or replace software, etc. to conform to new standards. In line with some of those new standards, I need to get you to reconfirm whether or not you are still interested in being notified by email about any new post to this blog. As part of the changes, all users must now re-subscribe, and those who do not reconfirm their interest will have to be erased from our records.

To reconfirm, fill out the form at https://thesilentknight.info/please-reconfirm-your-subscription/. Hope to hear from you soon. Thank you.

A Pair Of Perspectives On Pearls

Several Sundays ago, our pastor began his sermon by talking about the world’s largest pearl.

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This pearl, which was just recently revealed after being hidden for ten years, is 26 inches long and 12″ wide, weighing 75 pounds. That’s over 2½ times longer and five times heavier than the previous record-holder, which was found in another giant clam near the same Philippine island in 1934 (both the diver and the clam lost their lives in the earlier case).

Filipino Fisherman Reveals 75-Pound Pearl He Kept Hidden For A Decade.

Our preacher used the pearl as an example of how God can take an irritant and help you turn it into a treasure. I believe that is a common way of looking at this from a human perspective. But for some reason, I saw the issue differently.

Another Possible Moral To This Story

Turning trash into treasure – is that what the clam was actually trying to do? Does the clam even know that the old irritant now has such great value (estimated to be over a hundred million dollars)? Was this 75-pound object, which had been growing for over a hundred years^A, actually less irritating to the clam than the original grain of sand? I suspect not!

The preacher could have used this as an allegory, showing how man, because of pride, will try to solve a problem by himself but fail, despite putting a great deal of effort and time into it. And, as is often the case, he could even make the situation worse, not better. “But look how pretty I made it.” Can’t you just see that giant clam trying to sing Frank Sinatra’s “My Way”^{lyrics, video} (but failing, of course, because it has a 75-pound lump of calcium in its mouth)?

Is There A Third Possibility?

One of the beauties of life is that some situations can share many lessons. Reality is like that. Even as I was writing this, a third perspective began taking shape in my mind. Of course, I ignored it. But if you found another pearl of wisdom in this parable, please share in the comment section below. Thank you, and thanks for listening.

Truth In Advertising: An Unborn Baby’s Beating Heart

Above is one version of a common billboard along the interstates 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. 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.
The attrition rate of fertilized human eggs is not as well known. Comparing articles on spontaneous abortion in humans and Early embryo mortality in natural human reproduction, it looks like only 25% of fertilized eggs actually make it to birth (± at least 10%).

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 week^A1, while 2/3 of fertilized eggs will die in their first 11 weeks^A2.

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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%. That 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.

At least that’s the way I see it. Any comments?

Has the Republican Party Become a Ponzi or Pyramid Scheme?

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Definition of a pyramid scheme.

As a teacher, I didn’t heartily endorse “the Curve.” I didn’t believe that just because a student 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 (which turned into a prophesy). 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, then non-educated people could eventually reach critical mass. They could then elect enough unqualified politicians to negatively affect the education process (with budget cuts, for example). This 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 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?

From Your Crew: Why We Ask “How High” When You Tell Us To Jump

When bragging about their leadership skills, most of you have probably heard someone you know tell you “When I tell them to jump, they ask ‘How high?’”. You are supposed to assume that the jumpers do it to satisfy the most particular nature of the speaker’s demands. This implies that the speaker holds such power that their workers spend extra effort meeting even his or her most detailed or trivial requests. The speaker doesn’t expect you to wonder why their workers assume s/he has a need to micromanage every task. To keep your mind from wandering too far down that path, let me explain not the speaker’s mind, but go right to the true motives of their crew.

As a crew member, I know the speaker (let’s just call him or her “fearless leader”, or even “fearless” for short) has the financial resources and personal or political connections to hold his/her current position. But they got their leadership skills from studying Hollywood blockbusters or listening to other wannabees further up the chain of command. S/he hasn’t taken the time to learn the capabilities of their team. They don’t understand the complexity of the problem at hand. They really do like to bark orders, however. As a member of fearless’ crew, my motives are simple. I want to stroke his or her ego and get out of serious work. I may have been on my school track team, but if fearless is satisfied with me jumping four inches off the ground, that’s fine with me.

If fearless had a hangover, and I was asked to stand in for them one day, my crew would know not to ask frivolous questions. If one of them did ask “How high?”, I would be tempted to stop and have him or her jump as high as they could many times while other members of the crew documented the results. And if the last jump didn’t measure up? . . . .

My crew would know two things:

I expect them to give 100%. If he was on the track team, I would expect him to excel. I would have assigned it to someone else if the situation only demanded a 4″ jump. If the job demanded a spacecraft to be built, I’d give the job to my rocket scientist (which would probably be a different person than my track star).
I expect my crew to understand the nature and needs of the assignment in front of them and work it out. Whether the job demanded a 36-inch or a 24-inch high jump, I would expect my track star to be able to figure it out and complete the job without further guidance. At the same time, I expect my team members to know their capabilities and understand their limitations, as well as the other capabilities within the team so that if there is a mismatch, we, as a team, can effectively deal with it.

Is any of this too much to ask?

Not Quite Clear On The Concept “Innocent Until Proven Guilty”

Listening to discussions on Facebook about the Brett Kavanaugh nomination, I was surprised and disappointed to see the “innocent until proven guilty” principle so often misapplied (and in two different ways). It became clear to me that a lot of people just don’t understand the concept.

A Larger Doctrine

When something is being awarded to somebody, whether it is good, bad, large, or small, most people would like to think the recipient deserved the award. It is the presenter’s responsibility to make sure that’s the case. The more extreme the action, whether reward or punishment, the more effort the presenters should take to see that the award has been earned.

The Bad

When the award is a punishment, this doctrine takes the form of “innocent until proven guilty”. If the death penalty is under consideration, for example, we need to go to great lengths to be sure we aren’t making a mistake. I’ll save the discussion of the two types of possible errors – letting a murderer go free vs. hanging an innocent person – for another day.

The Good

When the action under consideration is a reward, one would expect some law of symmetry to apply, and it does. In this case, the slogan “innocent until proven guilty” has no place. Whether it is the Mega Millions jackpot or the Nobel Prize, one does not assume a prospect is ‘innocent’, or deserving, until proven otherwise. It is up to the claimant to prove they deserve the award. For the Mega Millions jackpot, that would be by showing the winning ticket and some form of identification. The Nobel Prize has even more stringent requirements. Republicans have no trouble applying this principle to welfare recipients but seem to get tripped up when it comes to Presidents and Supreme Court nominees.

The Ugly

In either of the above cases, it is well understood, as stated above, that the candidate will be fully vetted. And the more significant the award, the more serious the investigation. For someone to insist that a candidate is “innocent until proven guilty”, especially for a reward, and then refuse to hold a meaningful investigation into any evidence of guilt is the height of duplicity. But that seems to be the current state of the Republican Party. It hasn’t always been this way.

Anyone with a logical alternate interpretation of the facts is welcome to share. You can be sure the civility of this discussion will be maintained.

The 50% Rule – A New Proposal For Term Limits

Why Do We Need Term Limits

One problem in government is that the longer a politician is in power, the more out-of-touch with their district they tend to become, the less responsive they tend to be to their constituents, and the harder it is to remove them from office. In the pursuit of reelection, getting campaign contributions takes more and more of their time, and the focus on money rather than solutions to problems makes them more susceptible to less altruistic interests. As they gain funds and influence, even lousy incumbents have been shown to have a clear advantage in an election.

The simple answer has been to restrict each politician to two terms of office and then force ‘retirement’. I think that’s wasteful. A statesman requires critical qualities and talents that can improve with experience, and the qualities that separate the statesman from the politician seem to be becoming increasingly rare.

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It’s been said that “the difference between a politician and a statesman is that a politician thinks about the next election while the statesman thinks about the next generation”^E. For better definitions of ‘statesman’, check out the Houston Chronicle and/or “The Art of Manliness” blog (This is not an endorsement of the AoM blog; nor is this disclaimer a denunciation. I’m always suspicious when people define “real men” because it seems it’s usually the wannabees that speak loudest, and the result is usually watered down and/or distorted into something they can succeed at. Although I have not yet had the chance to investigate this particular site, this article is very good).

There should be a way to make the most of these skills without suffering the adverse side-effects.

A Better Plan – The 50% Rule

My idea would be to let a candidate serve as long as his constituents want him/her without ever letting them be the incumbent. This means after serving their term, they would return to their district for at least one term to reconnect with their constituents. They would not be allowed to hold any political office during that time, whether local, state, or federal.

It would also be good to have a separate law prohibiting them from working as a lobbyist for a set time longer than one term (maybe seven to ten years) after leaving office. And if not already on the books, a law should be put in place requiring a candidate to live in their district for at least one term of office before becoming a candidate. If the candidate is any good at being a statesman, s/he could end up with more than twenty years of public service. It would just take him/her over forty years to achieve that goal. The term incumbent and the advantages of being an incumbent would disappear while still allowing the gifted to fulfill a career in public service.

Other Helpful Ideas

Since we’ve mentioned the need to chase campaign funds as a root of this problem, it also makes sense to try to level the playing field there too. I believe the Citizens United decision by the Supreme Court was a mistake (see ‘If You Don’t Have To Pay For It, It Can’t Be Considered Free Speech” – Supreme Court). Steps should be taken to limit campaign contributions have the government fund viable campaigns, or even limit the duration of political campaigns to keep us from going from a one ‘man’, one vote system to one dollar, one vote. In debunking arguments favoring the electoral college (Another Look At The Electoral College), I also decry the ‘one acre, one vote’ notion. I discuss other possible improvements to the election process in Two Political Parties Are Not Enough.

As wise and principled as our forefathers were, even they could not imagine the changes in store in the future before them. That’s why they made provisions to change our rules of governing as needed. We may need to do that now to preserve the power that was originally intended for the people of this great country.

Teamwork Or Rugged Individualism

Almost two years ago, I was inspired to address this debate by a blogger that I very much admire (“Confident Individualism” by Lani Rodriguez). She presented a very nice case for the side of individualism, but now, echoing Joni Mitchell’s sentiments in “Both Sides, Now” (my second favorite song^blog), I’ve looked at individualism from both sides now. It may not be a simple this vs. that type of argument.

Part of growing up, or maturing, is being able to function on your own, meaning you need to make plans and take individual actions to solve problems as they arise, and then take the necessary follow-up to make sure those actions were effective. Some people never get this far, and others never get past this point. But there is a limit to how much one person can achieve alone.

As a member of society, the next logical step would be learning how to work with others – certainly not a trivial skill, but one that gives you the power to solve bigger problems, benefitting not just you, but your whole community. This too, as I’ve already implied, is a normal part of the maturation process; if you live in contact with others, as most of us do, this step is not optional.

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And if you can’t handle the team approach, I don’t even think you should consider having kids. “It takes a village . . .” after all.

So the title of this article, although written just as this question is usually presented, is deceptive.

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This is not the first discussion on this blog to address situations involving the misunderstanding of an ‘or’ statement^example. I have plans to discuss this further in a companion blog to The Problem With The ‘If’ Statement. Although it is pretty high on my topic list, I can’t say how soon it will be ready.

These are not opposing means to handle any task, but both need to be an integral part of your arsenal.

You could look at them as two gears in your vehicle to success. Individualism is first gear; teamwork is second gear. People complain that their upper gears are hard to maintain (“It’s too hard to get good help”, “It would be easier to do it myself than train someone else to do it”, . . .) and thus not worth the effort. It is true that the more moving parts, the more likely to break down, but your first gear is just as likely to fail you as any other. And even if it’s not, I’ve actually had a real car that couldn’t get out of first gear, and even though the repair facility was reasonably close, it took me forever to get there. Driving around in a car with only one gear is NOT an option unless you have really low expectations in life. So why didn’t I just call a tow truck? Oh, so now you’re arguing in favor of the team approach.