Awwww, Ray, I am so disappointed in you!
The clue you gave when you presented the puzzler was that another dimension was needed in order to solve the puzzle. Then when you gave the answer, you said your clue was incorrect.
The clue was correct! You cannot solve the puzzle without the 3rd dimension, height. In order to make the 16 right angles you MUST lift two sticks UP over the other two and move them towards the center. As soon as you do that you have created the 3rd dimension – however minuscule, it is still a 3rd dimension. Now you and I, being a bit older and suffering the consequential loss of good eyesight, may not be able to see something that tiny without magnifiers, but it still has height. — Ten lashes with Ann Landers’ wet noodle!
@LindaR, I think you are quibbling a bit here. For the term “angle” to make any sense, any two intersecting sticks which make an “angle” must be in the same plane. Ray is simply using the geometric projection of one stick upon the other to define the “angle” is all; i.e. Ray’s point is that there’s no need to use sticks that poke upward or downward to solve the problem.
@GeorgeSanJose The solution in three dimensions gives the same result (assuming each wooden match stick has a rectangular cross-section). Each intersection would still have 4 right angles, but in two different planes. The total number of right angles remains 16.