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You've probably all had some problem you've worked at.
And you keep going at it and you keep going at it...
Nothing seems to work.
The Einstellung effect, which is about
being fixed in a position;
these guys called Luchins and Luchins back, I think, in the 1940s
did experiments with water jars, and you've probably seen this sort of puzzle.
You've got a number of jars, and the idea is they are different sizes and you're trying to get
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– say – 2 pints of water and you've got
a 17-pint jug and a 3-pint jug, and you've got to pour water back and forth.
And there sometimes are really complicated solutions where you have to
fill one up from another, and then what's left in this one is some nice magic amount.
So, if you've got a 3-liter jug and a 7-liter jug and you fill the 7-liter one
and you pour it into the 3-liter jug and then throw the 3 liters away, you've got 4 liters in your 7-liter jug.
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And you have these complex solutions. So, what Luchins and Luchins did was they gave people
a number of examples which they worked through
which required that kind of complex solution, and then they gave them four more.
But the four more they gave them were ones where you could actually solve it very, very simply;
you know, possibly just filling up two and pouring them into another.
So, some subjects were given that, some were just given the four easy problems.
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The ones who were just given the four easy problems solved it in the easy way.
The ones who *started off with the complex ones applied the same complex heuristics*
to the *easier problems and took much longer to do it*.
So, they got so stuck in their ways having been introduced to this set of problems.
One that I encountered, myself – and this gives you an opportunity to laugh at me if you want to
because you'll probably, most of you I'm guessing might find this easy.
But I was in university and I was once given this puzzle.
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And the puzzle starts off, and there is this stage where you're driven down the wrong path.
Anyway, so I'm giving you a hint by telling you that.
So, you often get these puzzles where you're supposed to chop things up into identical pieces.
So, I've got a square up there, and obviously, you can chop a square into four identical smaller squares.
Or you can chop it into four triangles; so, the triangles are all the same shape as each other, the same size as each other.
In mathematical terms, they are *congruent* to each other.
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But they're not necessarily squares.
I've got some triangles, and we can chop the triangles into four – they're different kinds of triangles in different ways.
And on the right, I've got one of the ones you sometimes see as a trick puzzle in books,
or sometimes as you cut out pieces, as a sort of trivia-type puzzle.
And the idea there is to have four pieces that will make up that "L" shape.
And certainly, there may be other ways of doing it, but the way I've always seen is with
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these four pieces like that, so they're all "L" shapes themselves and they fit together.
How about cutting things into three?
Well, obviously the "L" shape's an easy one, and things that start off...
with three degrees of similarity like a triangle or a hexagon are relatively straightforward.
So, the crucial question is, can you chop a square into three pieces that are congruent with one another?
Not all squares, but some shape – they could be triangular, they could be more complex,
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that are each identical – can you do it?
Now, my guess is you've probably already thought of how to do it and you'll be right.
It took me three days to solve this because I was a mathematician
and the whole thing was framed in geometry, mathematics
and I was trying to apply really complex mathematics.
And, despite numerous hints, it took me three days to solve this.
If you haven't solved it, you're probably a budding mathematician.