Katherine and I are back in Seattle after a summer away, exploring the mathematics of  activities not normally associated with math: building a kitchen, planting gardens, and putting up goat fences. Of course, these activities were sometimes more mathematical than they looked. I used more trig than I have in years to make sure all the cuts on the miter saw were made at the correct angles.

Now we’re back, and beginning a new academic year. Our classes are off to a rollicking start, and we’re leading workshops for math teachers in schools around Seattle, which is a real pleasure. It’s sad to be away from the country, but good to be back in the city too.

In the spirit of our summertime activities, here are a few math puzzles to get the year started:

1. Squares have a way of becoming rhombuses if you’re not careful, and when you’re setting the walls onto the foundation, you have to make sure that your four perfectly-cut, 12-foot-long two by fours actually meet at right angles, forming a square, rather than skewing off and becoming a non-rectangular parallelogram. Fortunately, carpenters have a simple way to check that the wood is square by making just two measurements with a tape measure. Can you figure out how they do it?

2. Of all rectangles, the square is the most “efficient,” in the sense that if you have a certain amount of fence, and form it into a square, you’ll have a larger garden than if you form it into any other size rectangle. However, that fact goes out the window if you are fencing your garden against a preexisting wall. What rectangle is most efficient when you have the wall?

Have answers? Questions? Generalizations? Drop them in the comments and I’ll reveal my answers to these questions later.