Moving couch problem
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18 members have voted
August 2nd, 2017 at 7:52:32 PM
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This thread is along the theme of great unsolved math problems that can be phrased in plain simple English. This one is called the moving couch problem.
First, imagine you are living in Flatland (a two-dimensional world). In this world there is long straight hallway of width 1. Eventually, this hallway makes a 90-degree turn. It then goes on for another long distance. The width of the hallway is always 1.
The question is what is the maximum size couch (in area) that you can move down this hallway. Here is a picture of the hallway, just in case anyone is lost already. You need to move a couch from x to y. Think of every grid was 1x1.
You could obviously slide a 1x1 square down the hallway, around the turn, and down the other way.
You could even move a semicircle of radius 1 down the hallway, pivot it 90 degrees clockwise at the corner, and then move it down the other way. Now you're at pi/2 in area.
However, there are known larger sizes that work. The largest side known so far does not have an elegant shape but is about 20 separate pieces strung together in sort of a banana shape.
It seems to be there should be some elegant solution to the problem, but nobody has proven a maximum possible solution thus far.
The question for the poll is what are your thoughts so far?
First, imagine you are living in Flatland (a two-dimensional world). In this world there is long straight hallway of width 1. Eventually, this hallway makes a 90-degree turn. It then goes on for another long distance. The width of the hallway is always 1.
The question is what is the maximum size couch (in area) that you can move down this hallway. Here is a picture of the hallway, just in case anyone is lost already. You need to move a couch from x to y. Think of every grid was 1x1.
You could obviously slide a 1x1 square down the hallway, around the turn, and down the other way.
You could even move a semicircle of radius 1 down the hallway, pivot it 90 degrees clockwise at the corner, and then move it down the other way. Now you're at pi/2 in area.
However, there are known larger sizes that work. The largest side known so far does not have an elegant shape but is about 20 separate pieces strung together in sort of a banana shape.
It seems to be there should be some elegant solution to the problem, but nobody has proven a maximum possible solution thus far.
The question for the poll is what are your thoughts so far?
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
August 2nd, 2017 at 8:24:19 PM
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It feels like this problem could be solved with a computer simulation, why is that not the case?
In either any case, I voted for asbergers. When I was in Vegas, a pit boss asked me if I was retarded. I replied "autism is a wide spectrum".
In either any case, I voted for asbergers. When I was in Vegas, a pit boss asked me if I was retarded. I replied "autism is a wide spectrum".
August 2nd, 2017 at 8:31:27 PM
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Quote: gamerfreakIt feels like this problem could be solved with a computer simulation, why is that not the case?
In either any case, I voted for asbergers. When I was in Vegas, a pit boss asked me if I was retarded. I replied "autism is a wide spectrum".
Damn, if that's the first time someone's called you retarded, you've been running pretty good. I think I average twice a day.
As far as the question:
1) I assume it can't have pivots or whatever you call them.....like a snake that can move side to side.
2) ThatDonGuy probably gonna come in here and give the most elegant and simplest proof ever, something like, "Since pi + 1 = 1 + pi...." and it'll actually make sense.
EDIT: For #1, more like a hinge I guess, like a door that has a hinge on it. Can my couch have hinges??
August 2nd, 2017 at 8:33:43 PM
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I doubt that I will be able to focus my feeble brain on this one, and my initial thoughts are not very productive. They are:
(1) I would use an inflatable couch and deflate it for the move.
(2) At a minimum, I would use a flexible couch.
Sorry to be so unresponsive.
Edit: I was typing while RS was posting.
(1) I would use an inflatable couch and deflate it for the move.
(2) At a minimum, I would use a flexible couch.
Sorry to be so unresponsive.
Edit: I was typing while RS was posting.
August 2nd, 2017 at 8:57:37 PM
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Quote: gamerfreakIt feels like this problem could be solved with a computer simulation, why is that not the case?
Hmmm. I don't know too much about this, but I suspect human guided simulations brought us to the best known solution thus far. However, simulations are so unsatisfying, at least to me. Who can rest at night knowing the four-color map proof has allegedly been solved by computer? Not me.
Quote:In either any case, I voted for asbergers. When I was in Vegas, a pit boss asked me if I was retarded. I replied "autism is a wide spectrum".
Thanks. Best laugh I've had in at least a month. I'd love to hear the whole story.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
August 2nd, 2017 at 8:59:20 PM
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Quote: RSEDIT: For #1, more like a hinge I guess, like a door that has a hinge on it. Can my couch have hinges??
In response this and Doc's post -- no, absolutely no moving parts. The couch is as solid as a rock.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
August 2nd, 2017 at 9:18:01 PM
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I think you would need to know the floor to ceiling height of hallway. As well the width, depth, length, and padding density of said couch, in order to answer the question with any kind of certainty.
I vote to pay for delivery and let the movers deal with the problem.
How big is the apartment? My couch is inflatable.🤓😜
I vote to pay for delivery and let the movers deal with the problem.
How big is the apartment? My couch is inflatable.🤓😜
August 2nd, 2017 at 9:21:56 PM
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It's two dimension, no height -- no lifting or sliding. However, curved couches are allowed!
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August 2nd, 2017 at 9:33:40 PM
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I'm going with pi/2, a 1/2 circle shaped couch of width 2.
Edit: this is not the right answer.
Edit: this is not the right answer.
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You want the truth! You can't handle the truth!
August 2nd, 2017 at 9:53:35 PM
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I got a big couch...
that one end is touching the wall at point x, and the other is a molecule away from point y. The width of the couch is a molecule thinner from point x to the corner. By moving my big couch down a molecule, where it then touches y, I have accomplished the task of moving it from x to y.
Simplicity is the ultimate sophistication - Leonardo da Vinci
August 3rd, 2017 at 6:28:10 AM
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Quote: RSQuote: gamerfreakAs far as the question:
2) ThatDonGuy probably gonna come in here and give the most elegant and simplest proof ever, something like, "Since pi + 1 = 1 + pi...." and it'll actually make sense.
I heard that...
Actually, my first thought was that the problem was limited to "rectangular" couches. However, now that I see that the couch can be any two-dimensional shape, I have a feeling this is slightly above my mathematical expertise. I have a feeling the optimal shape is a portion of an ellipse/circle.
August 3rd, 2017 at 11:45:48 AM
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Quote: ThatDonGuyQuote: RS
I heard that...
Actually, my first thought was that the problem was limited to "rectangular" couches. However, now that I see that the couch can be any two-dimensional shape, I have a feeling this is slightly above my mathematical expertise. I have a feeling the optimal shape is a portion of an ellipse/circle.
Now I'm disappointed. :(
August 3rd, 2017 at 3:24:20 PM
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Not sure if this picture will work, if not consider a phone formed of two quarter circles (unit radius) with a join between them initially being a rectangle. The top of the rectangle joins the top of the quarters while the bottom just touches the corner at 45 degrees. Obviously a longer rectangle would be narrower, so there is an optimum size.
In addition to the rectangle there is another bit that could also be included - as the sofa rotates depending on the outside curve this defines what else can be added on the inside. I suspect it's a semi-circle for this shape.
Now by shaving bits here and there, one might be able to make the area larger with the rectangle bigger or the quarter circles slightly chubbier.
What seems obvious is that half way through the movement the shape round the corner is the same as the shape left to turn, so it must be symmetrical.
In addition to the rectangle there is another bit that could also be included - as the sofa rotates depending on the outside curve this defines what else can be added on the inside. I suspect it's a semi-circle for this shape.
Now by shaving bits here and there, one might be able to make the area larger with the rectangle bigger or the quarter circles slightly chubbier.
What seems obvious is that half way through the movement the shape round the corner is the same as the shape left to turn, so it must be symmetrical.
August 3rd, 2017 at 3:35:51 PM
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Quote: charliepatrickNot sure if this picture will work, if not consider a phone formed of two quarter circles (unit radius) with a join between them initially being a rectangle. The top of the rectangle joins the top of the quarters while the bottom just touches the corner at 45 degrees. Obviously a longer rectangle would be narrower, so there is an optimum size.
In addition to the rectangle there is another bit that could also be included - as the sofa rotates depending on the outside curve this defines what else can be added on the inside. I suspect it's a semi-circle for this shape.
Now by shaving bits here and there, one might be able to make the area larger with the rectangle bigger or the quarter circles slightly chubbier.
What seems obvious is that half way through the movement the shape round the corner is the same as the shape left to turn, so it must be symmetrical.
Couldn't those little triangles in the center of the "phone" be filled in?
August 3rd, 2017 at 3:49:50 PM
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Yes - as I said that adds even more. The objective here was to show the answer was more than the semi-circle solution. Where the outsides are circular, the inside will be circular (or in this case a semi-circle).Quote: gamerfreakQuote: charliepatrick....In addition to the rectangle there is another bit that could also be included - as the sofa rotates depending on the outside curve this defines what else can be added on the inside. I suspect it's a semi-circle for this shape....
Couldn't those little triangles in the center of the "phone" be filled in?
August 3rd, 2017 at 6:37:53 PM
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The more I think about this one, the more I believe it is a very interesting problem. Praise for Wizard for calling this to our attention.
Last edited by: gordonm888 on Aug 3, 2017
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
August 3rd, 2017 at 7:05:07 PM
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Quote: charliepatrick
Charlie, are you sure you could even get the couch into, and out of, that position? Remember that you have to move the couch around the corner.
However, your basic shape is not that far from the best-known solutions thus far.
The tough thing about this problem is that it is a moving problem. I know calculus is the mathematics of change but somehow I don't think it will help us with this. I can't think of any better solution than brute force trial and error. As soon as I have some more free time I plan to give this problem the ol' college try.
Following is in in-depth video on the problem.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
August 4th, 2017 at 4:53:33 AM
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I'm sure it couldn't! On reflection the bottom of the rectangle has to be in line with the points of contact, also at that point the inside shape should be a circle. Indeed it may show the outside curves are as well, but I've yet to analyze that. As you say there's lots of interesting reading about it although I may fall down when the integration comes into play! Thanks for posting the puzzle.Quote: Wizard...are you sure you could even get the couch into, and out of, that position?...
August 4th, 2017 at 8:24:00 AM
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This is one of those problems that I don't understand why it hasn't been solved. There's a known maximum size the couch can be, so why can't they just work backwards from that?
I've been accused of having Asperger's before. I think it's possible but so slight it doesn't really have much of a negative impact on my life. Aside from my terrible social skills, I guess.
I've been accused of having Asperger's before. I think it's possible but so slight it doesn't really have much of a negative impact on my life. Aside from my terrible social skills, I guess.
August 4th, 2017 at 10:01:06 AM
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Quote: TigerWuThere's a known maximum size the couch can be
erm. No there isn't. That is at the heart of the issue.
Psalm 25:16
Turn to me and be gracious to me, for I am lonely and afflicted.
Proverbs 18:2
A fool finds no satisfaction in trying to understand, for he would rather express his own opinion.
August 4th, 2017 at 10:14:57 AM
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Quote: OnceDearerm. No there isn't. That is at the heart of the issue.
This is what I mean:
Take a theoretical couch you know will fit around the corner. Add x amount of area. See if that still works. If it does, add x again. Keep going until you've reached a size that won't fit around the corner no matter how little area you've added or where you've added it. That is your maximum limit. I don't understand why that can't be done, if not by a person, then certainly by a computer.
August 4th, 2017 at 11:12:01 AM
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^ Say you started with the semi-cicrle and tried to add anything, you could prove you couldn't. That doesn't prove it's an absolute maximum - as has been shown. Thus adding a bit until you can't add anymore isn't good enough - it might be better to have a little less here so you can have more elsewhere. It's probably easy to work out a solution that is a local maximum (i.e. you can't add anything else), but that doesn't prove it's the maximum. The problem here is, there are so many ways to add a bit here and take a bit there, how can you prove there is no bigger solution?
There's a saying that you can prove something exists but it's harder to prove something particular doesn't exist. With the former you just have to find it, where, unless you can show some sort of contradiction, not finding one doesn't prove it doesn't exist.
There's a saying that you can prove something exists but it's harder to prove something particular doesn't exist. With the former you just have to find it, where, unless you can show some sort of contradiction, not finding one doesn't prove it doesn't exist.
August 4th, 2017 at 1:53:04 PM
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hmm, i took a shot at this and i got impossible answers.
i imagined a twinky shaped item, with a flat rectangular section in the middle and a quarter circle on each side. sort of like the phone, but the bottom is one constant line. a 4 sided shape with two of the sides being arcs, tangent to the top line. the middle section is of length X, and the arcs of radius R. a semicircle described in the problem statement would be this shape with X=0, R=1. i tried to attach a sketch, but that doesn't seem to be working for whatever reason.
i figured the 45 degree state was the critical section, so i put the 'couch' there at 45 degrees with the middle of the bottom line being right at the corner. going to the right, the hallway is a width of 1, so X/2*cos(45)+R = 1. the area of the couch is R^2*pi/2 + R*X = A.
by replacing X=(1-R)*2/cos(45) and taking the derivative, i find a maximum area of 1.59 sq units. pi/2 ~= 1.57 so maybe i've found a bigger couch. however i checked what values for R and X gave this geometry, it was for R=1.1245 and x = -.3521. this couch fits in the 45 degree position, but not in the flat section of the hallway. also it crosses over itself for some goofy areas. having a -X value is a little nonsensical.
i imagined a twinky shaped item, with a flat rectangular section in the middle and a quarter circle on each side. sort of like the phone, but the bottom is one constant line. a 4 sided shape with two of the sides being arcs, tangent to the top line. the middle section is of length X, and the arcs of radius R. a semicircle described in the problem statement would be this shape with X=0, R=1. i tried to attach a sketch, but that doesn't seem to be working for whatever reason.
i figured the 45 degree state was the critical section, so i put the 'couch' there at 45 degrees with the middle of the bottom line being right at the corner. going to the right, the hallway is a width of 1, so X/2*cos(45)+R = 1. the area of the couch is R^2*pi/2 + R*X = A.
by replacing X=(1-R)*2/cos(45) and taking the derivative, i find a maximum area of 1.59 sq units. pi/2 ~= 1.57 so maybe i've found a bigger couch. however i checked what values for R and X gave this geometry, it was for R=1.1245 and x = -.3521. this couch fits in the 45 degree position, but not in the flat section of the hallway. also it crosses over itself for some goofy areas. having a -X value is a little nonsensical.
August 4th, 2017 at 3:02:09 PM
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Here is what I came up with:
The two ends of the couch are Reuleaux triangles, which have a constant diameter.
I have no idea about other solutions, and I came up with this one playing around with my compass.
The two ends of the couch are Reuleaux triangles, which have a constant diameter.
I have no idea about other solutions, and I came up with this one playing around with my compass.
I heart Crystal Math.
August 4th, 2017 at 3:12:44 PM
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Hey, guys, I figured it out... If you have a bean bag couch you can make it infinitely long and it will still be able to squish around the corner.... :D
August 4th, 2017 at 3:27:22 PM
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Quote: TigerWuIf you have a bean bag couch you can make it infinitely long and it will still be able to squish around the corner.... :D
You're late.
August 4th, 2017 at 3:44:24 PM
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Quote: CrystalMathHere is what I came up with:...
Very nice. I can see how it would pivot around the corner. Best answer from this group so far. That gives us something to strive to beat at least.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
August 4th, 2017 at 3:50:13 PM
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August 4th, 2017 at 5:07:05 PM
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My second attempt was to bisect my couch and spread it apart while adding area between the pieces. I chose to spread them apart as far as possible, but then the connecting pieces would only touch at a point. Then I made some other realizations, as shown on the image:
Someone better at calculus or with more time than me could optimize the distance between my two original pieces (now quarter circles).
Never mind. The optimal distance is trivial. Center area = 2r- pi*r^2/2 . A simple derivative reveals the optimal radius is 2/pi.
Given that distance (as opposed to the one shown in the picture), the area is pi/2 + 4/pi - pi(2/pi)^2 / 2 = 2.2074.
Someone better at calculus or with more time than me could optimize the distance between my two original pieces (now quarter circles).
Never mind. The optimal distance is trivial. Center area = 2r- pi*r^2/2 . A simple derivative reveals the optimal radius is 2/pi.
Given that distance (as opposed to the one shown in the picture), the area is pi/2 + 4/pi - pi(2/pi)^2 / 2 = 2.2074.
I heart Crystal Math.
August 7th, 2017 at 3:31:05 PM
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I had one more thought about this. The idea was to take some area off of the top and extend the bottom, since the bottom seemed a bit wider. I created a calculation for the new area based on the distance d. Optimizing this distance, while keeping the center circle radius 2/pi,, I get d=.049077 and A = 2.2201684.
I don't see any reason why this wouldn't work, but I don't totally trust it because my area is greater than Gerver's Sofa. Although I don't entirely trust it, I did re-do the calculations twice and came to the same conclusion.
I don't see any reason why this wouldn't work, but I don't totally trust it because my area is greater than Gerver's Sofa. Although I don't entirely trust it, I did re-do the calculations twice and came to the same conclusion.
I heart Crystal Math.
August 7th, 2017 at 5:38:57 PM
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Maybe yours will become famous as the CrystalMath couch. Did you try cutting out the shape and moving it down a track with the requisite turn?
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
August 7th, 2017 at 6:29:52 PM
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I'm quite certain there is a small amount that needs to be removed. I was hoping to be done with this, since I'm obsessing about it.
I heart Crystal Math.
August 7th, 2017 at 7:00:26 PM
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Quote: CrystalMathI was hoping to be done with this, since I'm obsessing about it.
I would apologize but I don't want to lie to you. Who knows what kind of greatness is born out of such obsession.
I was obsessed with solving the Eternity II puzzle for months. Unfortunately, it turned out to be a huge waste of time. But I enjoyed the pursuit nonetheless.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
August 7th, 2017 at 7:17:28 PM
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CrystalMath: Maybe I am just so clueless that I don't understand how your latest solution makes the turn.
I think that in your 8/4 solution, as the couch moves along, there are two points each touching an inner wall, with the two ends of the couch slowly rotating about those sliding points.
If I have that correct, I don't understand how that translates to the latest solution. Instead of points, each end of the couch seems to be shown as having a short, flat, contact surface against an inner wall. I don't see how those flat surfaces will both rotate and maintain contact with the wall as the couch slides, the way that point contacts can. Perhaps the point/flat could be replaced with an appropriately curved surface to slide along the wall and rotate?
Am I missing something?
I think that in your 8/4 solution, as the couch moves along, there are two points each touching an inner wall, with the two ends of the couch slowly rotating about those sliding points.
If I have that correct, I don't understand how that translates to the latest solution. Instead of points, each end of the couch seems to be shown as having a short, flat, contact surface against an inner wall. I don't see how those flat surfaces will both rotate and maintain contact with the wall as the couch slides, the way that point contacts can. Perhaps the point/flat could be replaced with an appropriately curved surface to slide along the wall and rotate?
Am I missing something?
August 7th, 2017 at 8:06:38 PM
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Quote: DocCrystalMath: Maybe I am just so clueless that I don't understand how your latest solution makes the turn.
I think that in your 8/4 solution, as the couch moves along, there are two points each touching an inner wall, with the two ends of the couch slowly rotating about those sliding points.
If I have that correct, I don't understand how that translates to the latest solution. Instead of points, each end of the couch seems to be shown as having a short, flat, contact surface against an inner wall. I don't see how those flat surfaces will both rotate and maintain contact with the wall as the couch slides, the way that point contacts can. Perhaps the point/flat could be replaced with an appropriately curved surface to slide along the wall and rotate?
Am I missing something?
I think you got it right. The inner points need to be much more curved than I originally thought. I could have the same effect with the other points, but I'd need to remove much more area. Of course, it could be some balance between them.
I heart Crystal Math.
August 8th, 2017 at 2:52:14 PM
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Interesting problem - thanks Wiz!
It seems that the inner curve on CM's latest solution (the phone receiver shape) would be elliptical - not uniformly radiused - in the direction that reduces the area slightly. Just a gut feeling... I haven't started with arts and crafts yet!
It seems that the inner curve on CM's latest solution (the phone receiver shape) would be elliptical - not uniformly radiused - in the direction that reduces the area slightly. Just a gut feeling... I haven't started with arts and crafts yet!
It’s a dog eat dog world.
…Or maybe it’s the other way around!
August 8th, 2017 at 3:56:32 PM
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I'm thinking of buying a block of Styrofoam* and material to build a track just to see what I come up with. I would keep shaving the Styrofoam until it fit around the corner.
* Spell checker makes me capitalize that.
* Spell checker makes me capitalize that.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
August 8th, 2017 at 5:43:49 PM
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Quote: WizardI'm thinking of buying a block of Styrofoam* and material to build a track just to see what I come up with. I would keep shaving the Styrofoam until it fit around the corner.
* Spell checker makes me capitalize that.
It is trademarked by Dow Chemical.
https://en.m.wikipedia.org/wiki/Styrofoam
DUHHIIIIIIIII HEARD THAT!
August 8th, 2017 at 8:28:56 PM
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Quote: IbeatyouracesIt is trademarked by Dow Chemical.
https://en.m.wikipedia.org/wiki/Styrofoam
Sheesh. Do I have to capitalize Jacuzzi* as well?
* My spell checker says "yes." That has become an everyday word in the English language.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
August 8th, 2017 at 8:33:40 PM
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Quote: WizardSheesh. Do I have to capitalize Jacuzzi* as well?
* My spell checker says "yes." That has become an everyday word in the English language.
Sheldon Cooper
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You want the truth! You can't handle the truth!
August 8th, 2017 at 8:34:57 PM
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Dow's HQ is in Midland, MI. Next time I go to Soaring Eagle casino (we pass right by the HQ), I'll stop in and give them hell for you. 😉
DUHHIIIIIIIII HEARD THAT!
