Friday Puzzle -- more matchsticks

Hint

Alex Y

No takers on this one?

Note that the required figures referred to four-sided and five-sided areas, not to "rectangles" or "regular pentagons".

Re: Hint

Dan Donaldson

I sorta figured that, but for some reason, my head just won't wrap around this one.

Re: Hint

Alex Y

I'll post tonight or tomorrow morning. The four-sided one is surprisingly simple, but the five-sided one is harder.

Solution four-sided

Alex Y

For any given lengths of sides, a parallelogram can be drawn with any area from that of a rectangle to an arbitrarily small area.

While there are infinitely many solutions, one illustrating this principle is given by slanting the uprights of the original 2x3 rectangle as below:

The left-hand parallelogram has an area of 2 square matchsticks, while the original rectangle on the right has an area of 6.

Solution -- five-sided

Alex Y

The five-sided solution is more difficult. Both shapes are built up by combining equilateral triangles. I have shown the interior spots where matchsticks were used in laying out the solutions, but only the boundaries are left to count toward the 18-match total. The left-hand figure has an area of 5 triangles, while the right hand one has an area of 15 triangles.

Re: Solution -- five-sided

David Weaver

I majored in math, and I probably wouldn't have gotten those without several hours of exploring shapes and making a program to calculate their area for me.

I started to do the 4 sided on my sticky notes here at work, and then realized that it took me about 15 minutes to come up with one iteration and calculate the areas, and tapped out after that.

Re: Solution -- five-sided

Alex Y

I agree; I don't know that I would have ever gotten the five-sided one. (No one ever said that the poster was limited to puzzles he had personally solved!) Once you glom onto the idea of a parallelogram, the four-sided one becomes trivial.

Re: Solution -- five-sided

David Weaver

yes, trivial enough that it becomes possible to solve it even by taking more matches from the larger and giving to the smaller, and then just tiliting the parallelogram more. Could do it taking 2 or 4 more from it.

The five sided one becomes pure guessing, though!

Secretly, i'm hoping for more of these types, anyway.