Solving the "Impossible" Matchstick Triangles

[Treadstone 71]

In "The Equation That Couldn't Be Solved: How mathematical genius discovered the language of symmetry" by Mario Livio, he poses the following problem on page 268:

You are given six matches of equal length, and the objective is to use them to form exactly four triangles, in which all the sides of the four triangles are equal.

Now the "official" solution given in appendix 10 is to construct a tetrahedron. However, he claims that "the naive tendency is to attempt to solve the problem in two dimensions, where no solutions exist".

What about an equilateral triangle formed by 3 matches "cut" by 3 parallel, non-overlaping matches? The only reason why this solution would be false is due to the fact that the vertices are not at the end of the matches; however, this is not a requirement of the problem.

 

[Tide]

That will certainly work. I think the original statement of this classic problem requires that each match constitutes a complete side of a triangle - no more and no less. If Livio stated the problem as you indicated then he was being sloppy.

 

[Teegvin]

Your four triangles aren't congruent.

"in which all the sides of the four triangles are equal."

 

[Treadstone 71]

My interpretation of "all sides of the four triangles are equal" simply means that they are equilateral. The formulation of the question is somewhat sloppy.

 

[Tide]

Your four triangles aren't congruent.

"in which all the sides of the four triangles are equal."

They are congruent if the sides of the large equilateral triangle are bisected.

 

[Hurkyl]

However, he claims that "the naive tendency is to attempt to solve the problem in two dimensions, where no solutions exist".

I will disagree with this -- the tendency to solve the problem in two dimensions is because of the unspoken agreement that these types of problems are supposed to be done in two dimensions. Without this agreement, you would always have to explicitly state "two dimensions" when stating 99.9% of these kinds of problems. (and many, many more if you were against unspoken agreements in general)

 

[VietDao29]

...You are given six matches of equal length, and the objective is to use them to form exactly four triangles, in which all the sides of the four triangles are equal.
Now the "official" solution given in appendix 10 is to construct a tetrahedron. However, he claims that "the naive tendency is to attempt to solve the problem in two dimensions, where no solutions exist".

This reminds me of a story.
Last year, our teacher gave us that problem to do some thinking. At that time, we haven't learned 3D geometry yet (We have just covered it this year). My classmate, quite satisfied, proposed his answer as follow:
$$4 \triangle$$
Where the number 4 is made up by 3 matches.

 

[Treadstone 71]

Well, I must say, that is definitely thinking outside the box.

 

[Orefa]

Start with an equilateral triangle made of 3 matches. Cut it into 4 smaller but identical equilateral triangles using the remaining 3 matches: place each one at the midpoint of one edge and run it parallel to an adjacent side. Half of the top 3 matches stick outside the initial, larger triangle, but here again this is not forbidden by the problem statement. No need to go 3D.