For this week, I attempted to solve these two puzzles (pictured above) with my roommate Ellery. The first puzzled we tried is the blue one with the stars; the second was the multi-colored hexagon with triangles.

The first puzzle required us to connect all 16 stars without lifting our pencil off the paper, and by only using straight lines to go through the center of each star. Both of us tried looking for patterns in the stars- what kind of pattern could we make with 6 straight lines that would pass through all the stars?  In our mind's eye, we had to fill in mentally what we thought would be a good solution, because to keep tracing and retracing lines on the paper itself would've been confusing.  The process of solving the puzzle also included finding possible solutions through visual reasoning- "if I were to draw this line here, it wouldn't work, but perhaps..." etc.  In addition, we had to work with the available space and try to come up with a method to fill it.  In the end, we couldn't come up with the solution.

In the second puzzle we were asked to find one or more of the triangles that we were able to completely draw all the lines shown in the triangle without lifting our pencil from the paper, and without tracing any lines over.  Ellery got this one right away- she found the upper left hand triangle (the one made up of triangles) could be drawn as the directions instructed. Her reasoning was that she tried to find the simplest triangle first, and see if that would work.  In other words, she visually analyzed the figures she was given first, and then visually induced which one may give her a solution.  Also in this way, she categorized the shapes from easiest to hardest to draw in her mind. For me, I thought the triangle with the circle could somehow be a solution, but realized it would be impossible to not retrace any lines.  I also tried to categorize the shapes by what I thought would be easiest to draw, and I sought patterns within each shape.  There was actually another "solution triangle"--the one with the square in the middle.  But we didn't try finding it because we didn't realize there was more than one answer.