You were trying to put yourself into the same position as the students using the book, who didn’t already know pythagorean.
But you were not doing that.
You were systematically blocking knowledge you already had. That is very very very different from just not having that knowledge in the first place.
The thing that you were doing was hurting your ability to brainstorm and think about the problem at all. You can’t properly brainstorm and think of new, elegant solutions when you are trying to systematically block off a large part of your math knowledge.
The students who were using the book and didn’t already have that knowledge were not systematically blocking out knowledge. They just didn’t have it. So they are still able to freely brainstorm and think up solutions.
So you weren’t putting yourself in their shoes at all. You were doing something entirely different & much harder. The thing you were doing makes it impossible to think about things well. You didn’t even know what knowledge was required, what sorts of ideas were required, but you still blocked out a bunch of stuff. And knowledge is all connected, so in blocking out one thing, you don’t know what else you might also be blocking out.
And you weren’t just blocking out your knowledge of Pythagorean theorem.
So you were blocking out anything you thought would make the problem too easy.
So of course you had trouble finding the intended solution, which was actually elegant & simple.
Hmm. I think I was making some guesses about what sort of squares would be interesting or useful in some way and filtering for those. I guess that does sound like the sort of suppression you were talking about
To answer your question (hopefully!), in addition to the squares I mentioned above, there are 12 smaller squares in the picture - 8 squares consisting of a small triangle and trapezoid, and 4 black squares.
This is is a really common thing people do – in all areas not just math. I think it’s one of the major reasons they can’t learn philosophy.
One of the things it’s related to is looking for hard or complex solutions. This is partly social: they want to already be smart/clever/advanced, or to only talk about impressive things. But it has other reasons too, such as the belief that reality is complex. Goldratt discusses that in The Choice and argues for his view that reality is simple which he calls “inherent simplicity” IIRC. He explains that people often fail due to looking for complicated solutions instead of simple solutions.
Avoiding or suppressing simple stuff gets in the way of incremental progress. It blocks the building blocks that higher level concepts can be integrated from. It screws up the approach of practice and mastery. It screws up having a tree or pyramid of knowledge.
All the squares in Elliot’s picture!
22 squares in 6 different categories:
1 square consisting of the entire picture, whose formula is 4S + 16P + 16T = 1600
4 instances of the original square, whose formula is 1S + 4P + 4T = 400 4 squares whose formula is 1S + 3T + 3P 1 center square whose formula is 4T + 4P
4 black squares whose formula is 1S
8 smaller squares whose formula 1T + 1P
You’re adding in extra information based on having the solution. The formulas for this should be based on what you can tell initially from looking at the shapes.
I think a big reason people suppress brainstorming is they want to jump straight to the answer and not have or show any extra stuff, like false starts or reaching dead ends, or just generally having thought about anything that didn’t turn out directly useful.
This is how you’re sabotaging and blocking brainstorming.
The way I solved it didn’t even involve formulas or algebra, btw. You’re doing it in more complicated ways because you only ever responded to my prompts with with that stuff and refused to see the simpler stuff that is how I actually solved it.
And he also blocked out things he thought were too simple/easy, because he thought it was a hard problem. Like blocking out seeing(?)/mentioning the smallest squares.
So he blocked out both the simplest things and also the things he thought would simplify the problem and make it too easy, like the Pythagorean theorem. Blocking both simple and simplifying things covers almost everything.
You’re still missing key information from your formulas. I think you’re suppressing it. There are some facts that can be written in those formulas (by adding an equals sign and then something else that is the same area) which are easy to see by looking at the shapes.
One idea I have in my mind is that drawings can be deceptive, so I have a general hesitancy to say stuff definitively just based on the shapes.
But like, it would appear that the black square and the other smaller squares are the same size, so S = T + P.
It would also appear that the center square and the other “diagonal” squares are the same size, so S + 3T + 3P = 4T + 4P. Is that the kind of thing you had in mind?
Yes that’s it. One more is that you may notice the larger square is the size of 4 smaller squares, and you can get another equation from that.
You don’t have to get hung up on proving every detail. You can just try a lead and see if it works out. You can also go back and try to fill in gaps later that you made assumptions about, if you want to, like looking for a way to prove the white and black squares are actually the same size.
So if you write a list of all the equations you can then look at it and find 2 equations from the list which you can use to get a solution.
But, if you can’t tell that the white square & black squares are the same size, then you also can’t tell that any of the shapes you identified were squares. Like, calling the 2x2 squares “squares” assumes that the parallel lines are spaced evenly, or else they wouldn’t be squares?
You can’t even tell that the individual black & white tiles are squares, unless the lines are spaced evenly.
I don’t see you can tell that there are any squares in the picture at all (either the single tile white or black squares, or the larger 2x2 squares) if you can’t tell that the black & white squares are equal.
You seem to be referring the larger (2x2) squares as “diagonal” when the 1x1 squares are also diagonal. This shows you’re still blinding yourself to the existence of the 1x1 squares sometimes.
when you say “the larger square”, what do you have in mind?
This is my current revision
EDITED
1 square consisting of the entire picture, whose formula is 4S + 16P + 16T = 1600
4 instances of the original square, whose formula is S + 4P + 4T = 400 4 squares whose formula is S + 3T + 3P = 4T + 4P = 4S 1 center square whose formula is 4T + 4P = S + 3T + 3P = 4S
4 black squares whose formula is S = T + P
8 smaller squares whose formula is T + P = S
