A Nice Little Logic Problem

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djembeweaver
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A Nice Little Logic Problem

Post by djembeweaver » Wed Nov 11, 2015 4:11 pm

Three men are given the following task:

In an adjacent room, that is completely dark, there is a box containing 4 red hats and 2 blue hats. They must each go in one at a time, take a hat from the box, put it on their head then walk back into the (illuminated) first room again. Their task is to work out which colour hat they are wearing.

As the third man walks out of the room he sees that the other two are both wearing red hats and immediately declares that he is also wearing a red hat.

Using pure deduction, how does he come to that conclusion?

Assumptions:

The men know how many hats of each colour are in the box.

We must assume that if a deduction is possible then each man will make it...

No man can see his own hat.

This is a pure logic problem with no silly solutions like mirrors.
Last edited by djembeweaver on Wed Nov 11, 2015 10:53 pm, edited 1 time in total.

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Re: A Nice Little Logic Problem

Post by bobbo_the_Pragmatist » Wed Nov 11, 2015 6:22 pm

Using pure Aristotelian Logic, the only deduction that could be valid with the third guy seeing two other hats would be if the two other hats were BLUE. With two red hats showing, there is a 1in3 chance Third Guys hat is red, 2in3 that it is Blue.

Show us the Logic here, or NEVER post another logic puzzler....cause right now, I don't see how I could be wrong...............because................logic.

They guy could have red/blue color blindness? But you said no mirrors!!!
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Re: A Nice Little Logic Problem

Post by xouper » Wed Nov 11, 2015 7:53 pm


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Re: A Nice Little Logic Problem

Post by bobbo_the_Pragmatist » Wed Nov 11, 2015 8:13 pm

Really? Well..................this is going to be embarrassing. With the other logic puzzles I have failed, my spidey sense told me I didn't know what was relevant or that I was likely wrong, but I soldiered on.

On THIS ONE....the logic seems completely pure. Thanks for the links. I'll wait though...maybe a few sleep cycles will dream me the answer?
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Re: A Nice Little Logic Problem

Post by baggie » Wed Nov 11, 2015 10:02 pm

The problem is a little different than the answers submitted by xouper. In all these versions one must also stipulate that the men are clever (it relies on being able to assume that all the students can work out the others thinking). Also in the above version it is symmetrical so any of the men could have answered. I am also uncertain whether person three can really be sure he has a red hat because of this. The fact that he answers first suggests that he is quicker than one or two. But perhaps because the others are slower, they just haven't had time to work through all the possibilities, and three actually has a blue hat.

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Re: A Nice Little Logic Problem

Post by djembeweaver » Wed Nov 11, 2015 10:15 pm

bobbo_the_Pragmatist wrote:Using pure Aristotelian Logic, the only deduction that could be valid with the third guy seeing two other hats would be if the two other hats were BLUE. With two red hats showing, there is a 1in3 chance Third Guys hat is red, 2in3 that it is Blue.

Show us the Logic here, or NEVER post another logic puzzler....cause right now, I don't see how I could be wrong...............because................logic.

They guy could have red/blue color blindness? But you said no mirrors!!!


The answer is not based on probability bobbo. It's a pure deduction with a probability of 1 (given the assumptions).

I will certainly post the solution but I'd rather people tried to work it out. I can give hints...Besides, it's quite tricky to explain so the best way is to get there yourself with a bit of help...that's how I was introduced to the problem.

HINT: Imagine you are the third man. Start making some assumptions about what hat you mi9ght be wearing

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Re: A Nice Little Logic Problem

Post by TJrandom » Wed Nov 11, 2015 10:49 pm

Since there is no assumption that any of the men know that there are blue hats in the box, nor is there an assumption that the men must accurately guess the color of their hat - it is reasonable for the third man to assume that his hat is red too. Of course it could be any color, but that it is red is reasonable for him to assume.

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Re: A Nice Little Logic Problem

Post by djembeweaver » Wed Nov 11, 2015 10:50 pm

Btw I won't pretend that I found this easy to grasp...quite the reverse. I was first introduced to this at college. There were 6 or 7 of us at a table in a bar and I overheard one friend explaining it to another. After a while the second friend, in a sudden flash of insight, seemed to grasp the solution and proceeded to try and explain it to me. By this time the whole table was listening and every now and then someone would 'see' the solution in a flash of insight, then join the side that was trying to explain it. The really funny part though, was that every time someone successfully grasped the solution, and proceeded to try to explain it to someone else, somewhere along the explanation they seemed to lose the understanding. This happened again and again to the point that we concluded that there was only one lot of understanding at the table...so that every time someone understood, someone had to lose the understanding!

I think the reason for this phenomenon was that the solution requires several 'levels of logic' or 'layers of assumptions' such that one had to see the whole...the gestalt as one rather than as a sequential series of steps. This explains why, in the process of explaining each step, one lost the understanding of the whole. It also explains why the understanding always came in a flash of insight...like a light-bulb Eureka moment.

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Re: A Nice Little Logic Problem

Post by djembeweaver » Wed Nov 11, 2015 10:51 pm

TJrandom wrote:Since there is no assumption that any of the men know that there are blue hats in the box, nor is there an assumption that the men must accurately guess the color of their hat - it is reasonable for the third man to assume that his hat is red too. Of course it could be any color, but that it is red is reasonable for him to assume.


Bugger! SORRY!!!!!

They know how many hats of each colour there are!!!!!!!

SORRY!!!!

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Re: A Nice Little Logic Problem

Post by djembeweaver » Wed Nov 11, 2015 10:52 pm

And it should have been 4 red hats and two blue.

Sorry I totally messed this up because I did it from memory :(

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Re: A Nice Little Logic Problem

Post by djembeweaver » Wed Nov 11, 2015 10:54 pm

I've edited the original post.

Really sorry...feel like an idiot...it really is a great problem though...

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Re: A Nice Little Logic Problem

Post by TJrandom » Wed Nov 11, 2015 10:56 pm

djembeweaver wrote:
TJrandom wrote:Since there is no assumption that any of the men know that there are blue hats in the box, nor is there an assumption that the men must accurately guess the color of their hat - it is reasonable for the third man to assume that his hat is red too. Of course it could be any color, but that it is red is reasonable for him to assume.


Bugger! SORRY!!!!!

They know how many hats of each colour there are!!!!!!!

SORRY!!!!


OK, I am stumped...

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Re: A Nice Little Logic Problem

Post by djembeweaver » Wed Nov 11, 2015 10:57 pm

TJrandom wrote:Since there is no assumption that any of the men know that there are blue hats in the box, nor is there an assumption that the men must accurately guess the color of their hat - it is reasonable for the third man to assume that his hat is red too. Of course it could be any color, but that it is red is reasonable for him to assume.


And yes, they have to guess accurately.

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Re: A Nice Little Logic Problem

Post by djembeweaver » Wed Nov 11, 2015 10:57 pm

TJrandom wrote:
djembeweaver wrote:
TJrandom wrote:Since there is no assumption that any of the men know that there are blue hats in the box, nor is there an assumption that the men must accurately guess the color of their hat - it is reasonable for the third man to assume that his hat is red too. Of course it could be any color, but that it is red is reasonable for him to assume.


Bugger! SORRY!!!!!

They know how many hats of each colour there are!!!!!!!

SORRY!!!!


OK, I am stumped...


Do you want a hint?

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Re: A Nice Little Logic Problem

Post by TJrandom » Wed Nov 11, 2015 10:58 pm

djembeweaver wrote:
TJrandom wrote:
djembeweaver wrote:
TJrandom wrote:Since there is no assumption that any of the men know that there are blue hats in the box, nor is there an assumption that the men must accurately guess the color of their hat - it is reasonable for the third man to assume that his hat is red too. Of course it could be any color, but that it is red is reasonable for him to assume.


Bugger! SORRY!!!!!

They know how many hats of each colour there are!!!!!!!

SORRY!!!!


OK, I am stumped...


Do you want a hint?


Happy to wait........ maybe someone else will get it.

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Re: A Nice Little Logic Problem

Post by djembeweaver » Wed Nov 11, 2015 10:58 pm

TJrandom wrote:
djembeweaver wrote:
TJrandom wrote:Since there is no assumption that any of the men know that there are blue hats in the box, nor is there an assumption that the men must accurately guess the color of their hat - it is reasonable for the third man to assume that his hat is red too. Of course it could be any color, but that it is red is reasonable for him to assume.


Bugger! SORRY!!!!!

They know how many hats of each colour there are!!!!!!!

SORRY!!!!


OK, I am stumped...


Note that it should have been 4 red and 2 blue...:(

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Re: A Nice Little Logic Problem

Post by Gord » Thu Nov 12, 2015 2:13 am

I think I can work this out.

Wait, sorry, I'll put it in spoilers:

Spoiler:
His reaction must be to the lack of reactions of the other two, which must be because they don't know what colour their own hats are after seeing his.

So, if he was wearing a blue hat, then consider what one of the other men (let's pick man number two) would have been thinking. "The third man's hat is blue; if my hat were also blue, then man number one would know his hat couldn't be blue, so he'd say his hat was red. He hasn't done that, so my hat can't be blue, so I will say my hat is red."

So if the third man's hat had been blue, one of the other two would have spoken up and said his own hat was red. Since neither of them did that, man number three can't be wearing a blue hat, so he must be wearing a red hat.
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Re: A Nice Little Logic Problem

Post by bobbo_the_Pragmatist » Thu Nov 12, 2015 2:39 am

Gord--now that is interesting that "someone" in the room can figure out who is wearing what but the problem specifically says: "As the third man walks out of the room he sees that the other two are both wearing red hats and immediately declares that he is also wearing a red hat." So my spidey sense tells me if the challenge was worded a bit different the seeming impossibility could be staged....but not as framed. And I'm still thinking not with two red hats shown to the third man. Maybe one of the first two could declare something (with a different set up) but not the problem we have here.

Well, I hope there is a decent answer?
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Re: A Nice Little Logic Problem

Post by Gord » Thu Nov 12, 2015 3:05 am

I assume "immediately" means he had it figured out quickly, implying he was really really smart (like me! :nyanya: ).




...wait...it took me a while to work my way through it...so he must be...smarter...than me??... :blink: Is that even possible??
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Re: A Nice Little Logic Problem

Post by Gord » Thu Nov 12, 2015 3:06 am

I guess he could have known even without looking at the other two people if he had put three different hats on his own head. That way he'd immediately know that at least one of them was red.
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Re: A Nice Little Logic Problem

Post by bobbo_the_Pragmatist » Thu Nov 12, 2015 3:22 am

Gord....I feel you angst. It just hit me that much like Monty Hall, it shouldn't be too hard to make five hats with strips of paper and actually try the experiment on our desktop. Lots of these puzzle get clear real fast when you actually test them...rather than sit around and talk about them. Funny how true that is with so much of what we talk about...given that we do just about nothing?

.........................naw................I'm not going to do it. I enjoy the mystery too much.
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Re: A Nice Little Logic Problem

Post by TJrandom » Thu Nov 12, 2015 3:47 am

Gord wrote:I guess he could have known even without looking at the other two people if he had put three different hats on his own head. That way he'd immediately know that at least one of them was red.


That answer would be a `groaner`.... I do hope that isn`t it.

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Re: A Nice Little Logic Problem

Post by djembeweaver » Thu Nov 12, 2015 10:24 am

Gord wrote:I think I can work this out.

Wait, sorry, I'll put it in spoilers:

Spoiler:
His reaction must be to the lack of reactions of the other two, which must be because they don't know what colour their own hats are after seeing his.

So, if he was wearing a blue hat, then consider what one of the other men (let's pick man number two) would have been thinking. "The third man's hat is blue; if my hat were also blue, then man number one would know his hat couldn't be blue, so he'd say his hat was red. He hasn't done that, so my hat can't be blue, so I will say my hat is red."

So if the third man's hat had been blue, one of the other two would have spoken up and said his own hat was red. Since neither of them did that, man number three can't be wearing a blue hat, so he must be wearing a red hat.


Well done!

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Re: A Nice Little Logic Problem

Post by Gord » Thu Nov 12, 2015 10:32 am

Thanks for the puzzle. I like being able to stretch my brainwad open wide enough to hold everything needed to see the answer.

Or whatever that would be in non-Gordian English. ;)
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Re: A Nice Little Logic Problem

Post by djembeweaver » Thu Nov 12, 2015 4:51 pm

If you worked that out for yourself without looking it up then I'm very impressed...I had to be walked through it, though it seems blatantly obvious now.

Technically, according to the definition of the problem, all three men would declare that they are wearing red hats at exactly the same time...

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Re: A Nice Little Logic Problem

Post by xouper » Thu Nov 12, 2015 5:22 pm

djembeweaver wrote:If you worked that out for yourself without looking it up then I'm very impressed...

Yes, that is impressive. But I am not surprised, since Gord has demonstrated such mental abilities many times on this forum.

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Re: A Nice Little Logic Problem

Post by TJrandom » Thu Nov 12, 2015 7:15 pm

OK, so I looked at Gord`s spoiler. You couldn`t get that explaination across in a bar anytime after the first drink.

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Re: A Nice Little Logic Problem

Post by Gord » Thu Nov 12, 2015 9:11 pm

I remember solving a similar problem before, but I can't remember what it was. I had to work it out again from scratch. Juggling the "three people's individual interpretations" was very pleasurable for me. It reminded me of younger times, when I could look through a deck of cards one by one and eliminate them from an imaginary list in my mind, so that I would know what the last card was by deduction. I can't do that anymore....

Incidentally, someone in another thread mentioned Marilyn vos Savant, so I looked her up on Wikipedia. It mentions the Monty Hall problem, which we've been discussing recently, and the "two boys" or "second-sibling" problem, which is similar to the puzzle I posted long ago about the two dice problem. Serendipitous!
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Re: A Nice Little Logic Problem

Post by bobbo_the_Pragmatist » Thu Nov 12, 2015 11:21 pm

Dj==you cheated. What you "said" was: "As the third man walks out of the room he sees that the other two are both wearing red hats and immediately declares that he is also wearing a red hat.

Using pure deduction, how does he come to that conclusion?

Assumptions:

The men know how many hats of each colour are in the box.

We must assume that if a deduction is possible then each man will make it...

No man can see his own hat.

This is a pure logic problem with no silly solutions like mirrors.

I've highlighted the fraudulent terms you used to mislead us into your maze of mirrors. I'll allow the dictionary to be used to solve for how you set up the problem incorrectly with a simple change that would have fixed it all.

► Show Spoiler
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Re: A Nice Little Logic Problem

Post by Wordbird » Wed Jul 25, 2018 7:32 pm

Gord wrote:I assume "immediately" means he had it figured out quickly, implying he was really really smart (like me! :nyanya: ).




...wait...it took me a while to work my way through it...so he must be...smarter...than me??... :blink: Is that even possible??
So here goes. I'm going to address For clarity's sake, let's call the three men Moe, Larry, and Gord, Gord being the one who walks into the room and observes two layers of nonreaction and deduces that his hat is red.

Let's say Larry sees Gord come in with a blue hat. He looks to Moe, who does not react, and if Moe saw two blue hats, he would react, so Larry confidently declares that his hat is red.

Gord does not know whether Larry has said his own hat is red because of the reasoning in the spoiler, or because of the nonreaction of Moe alone.

For this problem to work if Gord's hat is blue, Gord must know that he is smarter than Larry, and that Larry would not make the second layer deduction based on second layer nonreactions. If Larry would make the second layer deduction, then Larry will say his own hat is red regardless of whether Gord's is blue or not.

Gord must know Larry is not capable of the second-layer reasoning for the juncture to work at which Gord considers, "If my hat was blue, Larry would say something."

But the problem is not claimed to work if Gord's hat is blue. Obviously it does not.

The problem, then, for Gord walking in with a red hat is that silence can only mean that both Larry and Moe are not as smart as Gord. Larry and Moe must be stupider than Gord for this bit to work, still.

The problem with that is that we now don't know how much stupider they are. We know we can't construct the problem so that all three men have equal intelligence, and we can't know the relative values from the nonreactions. Whichever one first cries that his hat is red might be the smartest, and reasoned all the way to the second layer, but he also might not be the smartest and have seen a blue hat, registered the nonreaction of the other red hat, and said his hat was red.

All we can know is that the time to make the base deduction (whether each man sees two blue hats or not), is less than the first-layer deduction (seeing one blue hat, looking to the other red hat for a nonreaction, and declaring his own hat is red), which is less than the second-layer deduction (observing two nonreactions of red-hats and making the connection in the spoiler), for each individual.

Gord must wait an appropriate amount of time for Larry and Moe to have certainly made the first-layer deductions, but not the second-layer. This entails a guess, which makes it not pure deductive logic anymore. Good thing he guessed right, and this is kind of an easy thing to get a feel for, especially if you can see where peoples' eyes are going, but it is still a guess.

Basically, TL;DR, I think Bobbo is sort of right, except that even with a better construction I don't think you can make this work purely deductively.

The weird thing is that everyone had a little piece of this truth, including the bit about there being only one lot of understanding at the table, and Gord for bringing up being smarter than others, since we provably can't construct this problem so the three men have equal intelligence.

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Re: A Nice Little Logic Problem

Post by bobbo_the_Pragmatist » Thu Jul 26, 2018 6:04 am

Well thanks Word. I reread my own post from only 2-3 years ago, and I don't get it. Not worth the time to reread even more to see if I can figure out what I said....so thanks. Hmmm....2-3 years? Can we deduce the slope of my decline?
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Re: A Nice Little Logic Problem

Post by Gord » Fri Jul 27, 2018 5:42 pm

Wordbird wrote:Basically, TL;DR, I think Bobbo is sort of right, except that even with a better construction I don't think you can make this work purely deductively.

The weird thing is that everyone had a little piece of this truth, including the bit about there being only one lot of understanding at the table, and Gord for bringing up being smarter than others, since we provably can't construct this problem so the three men have equal intelligence.
I took the following part of the original post to mean all three men would make all the deductions I could possibly make myself:
djembeweaver wrote:Assumptions:

...We must assume that if a deduction is possible then each man will make it....
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Re: A Nice Little Logic Problem

Post by Wordbird » Sun Jul 29, 2018 2:42 am

Gord wrote:I took the following part of the original post to mean all three men would make all the deductions I could possibly make myself:
"...We must assume that if a deduction is possible then each man will make it...."
Right, they will, but you can't construct the problem so that they will make it as quickly as the other fellows.

Otherwise they'd all blurt out any possible deduction at the same time, and that didn't happen.

And it clearly matters for the problem that the man who eventually says his hat is red must know roughly when a failure to react means the other two notice each other's non-reactions on a particular reasoning level.

In other words, Gord has to be smarter than they are - able to make the deduction more quickly. So how do we now know Larry and Moe weren't still thinking?

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Re: A Nice Little Logic Problem

Post by Gord » Sun Jul 29, 2018 6:51 pm

My assumption was that they would make the all the deductions I could possibly make myself, just as I would have made them -- therefore in a timely manner equivalent to my own timing. Or maybe I could call that a presumption rather than an assumption. But I'm not sure about that.
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Re: A Nice Little Logic Problem

Post by landrew » Sun Jul 29, 2018 7:07 pm

Gord wrote:My assumption was that they would make the all the deductions I could possibly make myself, just as I would have made them -- therefore in a timely manner equivalent to my own timing. Or maybe I could call that a presumption rather than an assumption. But I'm not sure about that.
Try some nice soothing chamomile tea.
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Gord
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Re: A Nice Little Logic Problem

Post by Gord » Sun Jul 29, 2018 7:28 pm

landrew wrote:
Gord wrote:My assumption was that they would make the all the deductions I could possibly make myself, just as I would have made them -- therefore in a timely manner equivalent to my own timing. Or maybe I could call that a presumption rather than an assumption. But I'm not sure about that.
Try some nice soothing chamomile tea.
What, instead of my coffee enemas??? Don't be ridiculous!! Chamomile tea does go up there....
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Re: A Nice Little Logic Problem

Post by Wordbird » Mon Jul 30, 2018 4:18 pm

Gord wrote:My assumption was that they would make the all the deductions I could possibly make myself, just as I would have made them -- therefore in a timely manner equivalent to my own timing. Or maybe I could call that a presumption rather than an assumption. But I'm not sure about that.
If we construct the problem this way, no man can be the first to cry that he has a red hat. All three men (seeing two other red hats) would either make the deduction simultaneously, or not at all.

Besides, according to the logic in the problem, only the first man will get a correct answer.

As soon as Gord cries that his hat is red, Larry will look at the two red hats he sees and cry that his own hat is blue. He will assume that Gord saw Moe's red hat, Larry's blue hat, and Moe's non-reaction (Moe not seeing two blue hats).

In short, here's the problem: We're supposed to accept that when seeing two red hats yourself, a reaction could mean you have a blue hat, or a red hat. But that nonreaction means you have a red. The problem then foils that because it supposedly proves how not only the first man, but all three, could react if your hat is red. Therefore you can't deduce anything from reaction, or nonreaction. In either case your hat could be red, or blue.

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Re: A Nice Little Logic Problem

Post by bobbo_the_Pragmatist » Mon Jul 30, 2018 5:06 pm

Good Notions....I agree. Hats Off...….to you. ((Find relevant smilie and post here!))
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