Math Fun

[tab]Due to the symmetry, it doesn’t matter if you begin with rows or columns, so let’s choose rows.
In the top row, there are an odd number of spaces to fill thus
There is no alternative but to share a domino between the first row and the second.
The domino could be placed on either the right or left side of the board but in either case
There is no alternative but to cause an odd number of spaces on the same side of our domino as the missing space was on
We now have the same situation of even on one side an odd on the other but on the second row thus
We have no alternative but to share a domino with the third row and restricted to the odd side which
Leads to the same situation with the fourth row with the odd number shifted opposite as the last
As we go from row to row, we cause the next row to be odd on opposite side of our domino depending on how we started
That pattern has no alternative but to continue down the rows. With each row, the odd number of spaces shifts from right to left.

In order to solve the problem, because the last row has a missing space opposite from the first row,
We must have the odd number of spaces shifted to the opposite side from where we began with the second row
But there are an odd number of rows from the second to the last yielding no alternative but to have
the odd side be the same on the last row as on the second row.

Thus there is no solution.[/tab]

You asked me the exact same thing before. I then offered three posts, none of which got a response from you. You later told me you didn’t understand them. I haven’t simplified the logic, so I can’t help you there.

Where did you demonstrate false steps in James reasoning? And did James concur that he stood corrected? I must have missed all that, but given James compliant attitude now perhaps it did happen.

Those posts were just rambling. And as I said, general claims about hypotheticals are not a demonstration.

I demonstrated the false steps in his reasoning in that post of mine in which I made some phrases blue and some red. James need not concur: he has not shown how his reasoning is impervious to my critique.

Yeah… whatever you say I guess… lol.
It took you two years and a very intensive private workshop to see the logic to value ontology. Before that time it had been “just rambling”. This is a bit predictable.

Personally I think it’s ridiculous to say that your demonstration of “why it is unsolvable” is invalid as soon as the squares are coloured differently to a chessboard (with 2 opposite corners removed). As you say, if all squares are the same colour, they can of course still be treated as though they were coloured like a chessboard.

This talk of a generalised solution is interesting. Here’s a start, perhaps (not sure if it really needs to be tabbed anymore):
[tab]To advance the observation that, starting along any given side, because each side has an odd number of squares, at least one 2x1 rectangular tile would have to protrude into the next row/colomn, and that this would necessarily also give the next row/colomn (at most) 7 empty spaces like the previous one:

It is important to note that ONLY an odd number of 2x1 rectangular tiles can ever protrude into the next row/colomn, from the previous one.
E.g. if all 2x1 rects protrude into the next row/colomn, there are 7 of them. If all but 1 protude into the next row/colomn, there are 5 that do protrude (a 2x1 rect turned 90 degrees from its protruding state obviously covers 2 squares), if all but 2 protrude, there are 3 that do protrude, and if all but 3 protrude, there is only 1 that protrudes.
As already noted, this pattern will continue all the way to the other side of the board.

Now consider a property of the 8x8 board (with 2 opposite corners removed): it has ROTATIONAL symmetry when empty of 2x1 rects.
This means that if we rotate any 4x8 half (with 1 corner missing), it will map over onto the other half.

So now imagine we only advance this “odd number of protruding rects” pattern only half way. In any given combination, there will be an odd number of protruding rects, when filled so that there are no gaps, only protrusions.

Consider now that an odd number of protruding 2x1 rects from one half CANNOT lock into the other half because of this rotational symmetry, which requires that each protrusion MUST have a matching protrusion, an equal number of squares from the middle, which could be removed to let the one half lock into the other. Rotational symmetry requires that (on an even x even square board) there must be an even number of protrusions from either half. There can only be an odd number.

Writing this, I realise that each half doesn’t have to be symmetrical with the other, but somehow I can intuit that this isn’t actually a proper problem to this starting point.[/tab]
As for the “don’t remove two squares that would have the same colour if coloured like a chessboard, remove two that would have opposite colours” thing:
[tab]A simple algorithm that resembles “laying bricks”, with 2x1 rects parallel to the base of the board whenever possible (only “upright” when necessary), and completing one layer before advancing to the one on top, will solve any 8x8 grid with 2 squares removed that would be of opposite colour if coloured like a chessboard.

I dunno how to explain that in proof form, but it just works.[/tab]

Might I suggest you try explaining them again, but in an improved way?

Perhaps if they were not clear enough to merit a meaningful criticism the first time around, you ought to re-write it in a clearer way. I vaguely remember not understanding what you were getting at either, though perhaps at least offering the criticism that you cannot judge “what a smaller number of islanders would do if there were that smaller number of them” whilst artificially attributing knowledge to them that which only a larger number of islanders would know. I seem to remember that inconsistency in method applying to at least some of what you have offered.

I am very used to people coming around to my ideas a couple of years after I present them. Almost without exception, such ideas are at first arrogantly dismissed (without proper argument) by the people who later come to embrace them, even though by then they’ve often forgotten where they got them. So until someone responds to what my points rationally and to the point (I am hoping Carleas will in a promised PM) I am not quite exactly prompted to think that they’ve been understood.

Your posts were just rambling—i.e., all over the place. And indeed, why must you refer to three posts?.. Kindly give a systematic account of your reasoning, in one post. You can copy/paste from those three posts if you think it’s functional (I doubt it).

By the way, I still don’t think value ontology is the revolutionary innovation you so arrogantly proclaimed it was.

I think so too. It’s a great explanation making use of a valid example-form. I very much enjoy this sort of insight.

Yeah… I think you’re just way too lazy on this one. As I said I’m used to people taking a lot of time coming around to my logic, which admittedly gets very involved and counterintuitive.
The fact that y’all copied the canonical and began defending it at a point where it wasn’t even being attacked, but all failed, even refused to demonstrate the epistemic necessity of the guru (which is the point you’re relying on), has gradually led me to think perhaps you didn’t really penetrate the logic of that solution.

I did penetrate that, first by thinking that the hypothesis “if there was only one and the guru said she said one” was false. And it still is. But that doesn’t matter, as it is meant to be false. Its falseness is its epistemic status. Start there, do you understand what I mean when I say that?

This is a puzzle about epistemology, about the status of knowledge in epistemic agents. On the surface, the guru is a great tool. But the structure is not as simple as the three of you make it out to be.
By the by, as I’ve shown (and as Sauwelios even admitted when I told him “humbly” (the key to teaching) the explicit text doesn’t even allow for the canonical solution, as all people see each other at all times, disallowing for the whole “in the morning they were still there” scenario. But that’s just a minor flaw, should not distract from the major falseness of the first hypothesis.

I was right then, you’ve contributed exactly nothing. And indeed, to nothing, no one need concur.

Seriously, all I get from you is “I don’t understand, and that’s your fault”.
But it’s not only my fault - I make things too complex for you, you want things to be more simple than I can make them.

This is just an ungrounded insinuation. Are you in fact trolling here?

The “if” part of an “if/then” statement (jargon?) cannot be true or false, as it’s not a complete statement… Anyway, your reasoning seems to go like this:

  1. The one, blue-eyed person does not know his own eye colour.
  2. The guru tells him that she sees a blue-eyed person.
  3. The one, blue-eyed person, supposing that the guru is to be believed, knows he has blue eyes.

Now according to you, it seems, the 3rd step somehow retroactively cancels the 1st. This is nonsense.

I didn’t “admit” that, it’s just a truism.

Perhaps it’s because you don’t think in words, that you can’t put your thoughts into words very well.

Yeah… I think you’re just getting carried away on a cloud of self-believed but undemonstrated superiority on this one.

The fact that you’re attacking a strawman just leads me to think you didn’t really penetrate the logic of my/our solutions.

See how when I emulate how you speak, it sounds annoying? Yeah, that’s how you’re sounding when you speak. I recommend you follow your own advice and switch back to humble mode if you at least expect your theory on teaching attitudes to be valid - else it becomes apparent that you don’t (and perhaps never did) intend to teach anything, just say stuff and “proclaim” it’s correct, but just too deep for us to get (that JSS-style tactic won’t fly I’m afraid).

Please start by explaining why the Guru “saying she sees one” is incompatible with the thought experiment of “what one would do if there was only one”.

As for the “all people see each other at all times” thing - this is not incompatible with anyone leaving or staying on the island (perhaps the ferry doesn’t take them out of view, and they might even see perfectly well at night and never sleep for all we know). The whole “see them in the morning” thing is perfectly fine when taken as it was intended, as “seen on the island even after the ferry had been and gone, picking up anyone who knew their eye colour while it was there at midnight - as in the puzzle”. Even this attempt at a minor point seems to hold no water.

And please, to both you and Sau, stop squabbling. It really doesn’t lend yourself any of the credibility either of you are fighting for here.

It’s my honest impression. Should I accuse you of trolling every time you fail to be convinced by some point I want to make?

It is falsely inserted.

“the blue eyed person does not know his eye color”
“the blue eyed person does know his eye color”

Yes, these statements contradict each other.

You (all) seem to think that “Supposing that the guru [speaks and] is to be believed” applies only to step 3. Where, if it is valid at 3., it would also have been valid at 1. which cancels out the content of 1.

Oh wait, I am thinking.
I mean rambling.

It’s simply true. But not to the point here.

Abstract thoughts, when they are in fact a force, insight, always are hard to put into words. Clever word-play is often mistaken for thinking. I’m not of that school, indeed.

I personally doubt that there isn’t. A proof of a biconditional like “the problem is solvable IFF the board has [some property or set of properties]” would necessarily prove both.

On a much smaller board, the mutual solution is easy: A two by two board is solvable if and only if the removed squares are adjacent, because that implies that the remaining squares are adjacent, and the remaining 2-square board is solvable if and only if it is a 2x1 rectangle.

I would guess that adjacency is a special case of some property for which color is a proxy, and that “the board is solvable IFF the removed squares have property X such that X is a superset of adjacency”.

Oh really?
I’m listening…

“IF they all start counting from the same number, will they always be able to deduce their eye color?”

I found that much less annoying than how you normally write in this thread. It seems at least far less arrogant and oblivious.

Jesus, just fucking read my posts. My God. Or better yet, think.

Right. Sau says it’s a truism, you say it’s false -
you guys are clowns.

Sillybilly you are a true comedian.

This does not suggest that either is wrong, but that the problem as originally stated is ambiguous. I think your clarification, that the islanders can only see each other during the day, and they disappear during the night if they learn their eye color, resolves the ambiguity.

Right. So saying “just fucking read my posts. My God. Or better yet, think” as though your posts are so obvious (even though nobody gets them), and that I’m the one who isn’t thinking even though what I’ve said has been agreed with on multiple occasions, and even referenced in order to help combat the two trailing contributors who still don’t get it.

What you say doesn’t make any more sense now than it did the first time I read it. Take a leaf out of my book - from the part where I have proactively improved on my original presentation of the correct solution on multiple occasions, just to more clearly rebuke all the presented criticisms of it and how it is the only correct solution. That’s how you gain credibility: proactively showing and improving on the many hours of thought you’ve put into solving a problem and problems with others not understanding it. And when they don’t, try again and harder until they do.

Like I said: your lazy approach of merely claiming correctness without (clear) demonstration - whilst only angrily pointing fingers at others for not being as clever as you think you are “to back you up” will get you nowhere.

Er, have the 9-11 attacks cancelled out the fact that the Twin Towers ever existed?