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# 4 Identical Triangles

47 replies to this topic

### #21 Gerardo

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Posted 19 October 2007 - 04:43 AM

you can do the diamond form, moving 2 stick

/_ /_

'''''''_
/ /_

_
X

that's why i think, it doesn't have a solution, or is the wrong question

excuse my english, i need to practice
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### #22 Penguin

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Posted 30 October 2007 - 03:12 PM

Despite there existing a joke solution, there also exists a real solution.

```/_/_

/_/_```

That crude ASCII drawing of it might not make it clear, but if you draw it out yourself you'll see. If you just take one of the outside sticks and put it below, at the same angle, such that its top end is touching the point where the original triangles meet, then you have your solution: 4 identical triangles in 1 move.

I think people were just getting stuck because they assumed that a triangle must have three lines. But if you took basic Geometry, you should know that if you know the size of one angle and the length of two sides, then you can figure out the other two angles and the other side. Now, assuming that the matchsticks are all of equal length (which they are in the original image for the puzzle), then we know that in fact all the angles are 60 degrees, because this is true of any triangle whose sides are all of equal length (the angles of any triangle must add up to 180 degrees, and if all the sides are equal then all the angles must be equal, so 60 * 3 = 180 is the only possibility). Therefore, if we preserve the same orientation of the matchstick when we move it, then the new angle will also be 60 degrees. From here, it's a rather simple matter to mathematically prove that all four triangles are in fact identical.
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### #23 \$NIPEHUNT

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Posted 09 November 2007 - 09:05 PM

Ok, I've got the solution.

matchstick.JPG[/attachment:12be6]

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### #24 mindwarp

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Posted 12 November 2007 - 11:22 PM

Penguin wrote:

Despite there existing a joke solution, there also exists a real solution.

```/_/_

/_/_```

I think people were just getting stuck because they assumed that a triangle must have three lines. But if you took basic Geometry, you should know that if you know the size of one angle and the length of two sides, then you can figure out the other two angles and the other side.

I had considered this solution as well, however, I have to disagree with Penguin's line of logic. A quick query at dictionary.com reveals that a triangle is "a closed plane figure having three sides and three angles."

Each of Penguin's proposed triangles have only two sides and one angle, which is actually defined as an angle, not a triangle.

If you took basic geometry, you would probably also remember that when proving triangle properties, you actually need a triangle to prove. I don't recall any proofs that took an angle, added an arbitrary third side, called it a triangle and proved its properties. Granted we know the length of two sides, and the angle can be inferred from fact that the original configuration of the matchsticks suggest the triangles are equilateral, we have no proof what the length of the third side would be. Although we can assume (if you took geometry, you might have learned not to make assumptions) that the third side would be composed of matchsticks of the same size, we cannot prove this. This would require the addition of three matchsticks into the problem, which the original statement didn't mention. Again, we could assume that because the original problem didn't state "and you can't add any extra matchsticks," I think I mean to drive home the point how improper, logically, assumptions are.

Thus, Penguin's solution only results in one triangle.
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### #25 Penguin

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Posted 13 November 2007 - 03:16 PM

Penguin wrote:

Despite there existing a joke solution, there also exists a real solution.

```/_/_

/_/_```

I think people were just getting stuck because they assumed that a triangle must have three lines. But if you took basic Geometry, you should know that if you know the size of one angle and the length of two sides, then you can figure out the other two angles and the other side.

I had considered this solution as well, however, I have to disagree with Penguin's line of logic. A quick query at dictionary.com reveals that a triangle is "a closed plane figure having three sides and three angles."

Each of Penguin's proposed triangles have only two sides and one angle, which is actually defined as an angle, not a triangle.

If you took basic geometry, you would probably also remember that when proving triangle properties, you actually need a triangle to prove. I don't recall any proofs that took an angle, added an arbitrary third side, called it a triangle and proved its properties. Granted we know the length of two sides, and the angle can be inferred from fact that the original configuration of the matchsticks suggest the triangles are equilateral, we have no proof what the length of the third side would be. Although we can assume (if you took geometry, you might have learned not to make assumptions) that the third side would be composed of matchsticks of the same size, we cannot prove this. This would require the addition of three matchsticks into the problem, which the original statement didn't mention. Again, we could assume that because the original problem didn't state "and you can't add any extra matchsticks," I think I mean to drive home the point how improper, logically, assumptions are.

Thus, Penguin's solution only results in one triangle.

Well, you made two mistakes here.

The first is that you claim you cannot prove the length of the third side, but in fact you can, as I have already explained, but I will reiterate for you. We know the angle is 60 degrees, and we know each side is the same length. We do not know the actual length of each matchstick, but we know that they are all of equal length (which you can tell from the original problem's image). However, it is entirely irrelevant what their actual length is. Suppose that they were 10 (of your favorite units). Thus, since both sides are 10, and your angle is 60, then the ONLY possible length for the remaining side is 10. Try the math out yourself if you don't believe me. Then change the matchstick size to 100 or any other number you want, and it still comes out the same. We know the sides are equal, and the angle is 60, thus we know it is an equilateral triangle and the remaining side can only be the same length as the other two sides.

Also, I'd like to mention that your calling it an "angle" is incorrect, because an angle has two sides of arbitrary length, while we know for a fact that these sides are of finite length, and in fact the same length.

Your second mistake is your taking this concept of a "triangle" far too literally. Here is a helpful fact for you: There is no such thing as a "triangle". A "triangle" is simply a name given to a particular "shape". A "shape" does not exist in reality, but exists only in our minds. It is merely a pattern of light that we have evolved to recognize because it helped us survive. Take any three "points" (i.e. a visible mark whose location can be specified by a single pair of coordinates, or a three-tuple if you're dealing in 3 dimensions) and you have yourself a triangle. However, if you don't want to see it as a triangle, then you don't have a triangle. Like I said, it's all in your head. The concepts of shapes, colors, and distinct physical objects are all just abstract simplifications in our minds. If we went around seeing what's really there, we'd go insane. If you are interested in what you're "really" looking at, then I suggest you study quantum physics.

Anyway, my point is, just because a third line is not drawn in front of you doesn't mean that you are not looking at a triangle. A triangle is just a pattern, and a pattern exists as long as you have enough information to prove that it exists, which we do. In fact, all you need to prove the existance of a triangle is three points. If you have three points, then you have a triangle. And we do have three points: One where the two matchsticks meet, one at the end of one matchstick and another at the end of the other matchstick.

Now, you quoted me this definition: "a closed plane figure having three sides and three angles." If you truly understood math, you would know that this is not a definition of an actual entity existing in the real world, but rather an abstract concept that makes it easier for humans to make calculations. I believe Einstein explained it best: "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."
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### #26 Linzd21

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Posted 15 November 2007 - 03:55 PM

I understand the "trick" behind the puzzle. However, the title needs to be changed to "4 triangles" instead of "4 identical triangles. Because it doesn't matter how you explain the joke...in order for something to be identical to something there has to be more than one of it. there is no way to get 4 [u]identical[u] triangles from moving one matchstick in this puzzle.
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### #27 Penguin

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Posted 15 November 2007 - 04:32 PM

I understand the "trick" behind the puzzle. However, the title needs to
be changed to "4 triangles" instead of "4 identical triangles. Because
it doesn't matter how you explain the joke...in order for something to
be identical to something there has to be more than one of it. there is
no way to get 4 [u]identical[u] triangles from moving one matchstick in
this puzzle.

1: I've already demonstrated a non-joke solution that satisfies the 4 identical triangle requirement.
2: Your reasoning is also flawed, because x = x. In other words, any given thing is identical to itself.
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### #28 jenbarnes73

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Posted 16 November 2007 - 03:00 PM

The solution is to make a 3 dimensional pyramid. There will be 4 identical triangles with the same amount of sticks.
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### #29 Penguin

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Posted 16 November 2007 - 03:05 PM

The solution is to make a 3 dimensional pyramid. There will be 4 identical triangles with the same amount of sticks.

Well, the problem statement didn't limit itself to 2 dimensions, but I don't see a way to do such a thing with only a single matchstick move.
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### #30 jenbarnes73

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Posted 16 November 2007 - 03:16 PM

This is a pretty old teaser, it should have just said to make 4 identical triangles with the same amount of matchsticks.
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