Numberplay: Heights of a Triangle Here's how it works. Each week we take a simple math puzzle, solve it, and then explore the standard solution. Did the solution really work? Are there other solutions that would be just as good - or even better? Reader Caleb put it well: At first I thought the question was too easy, but the real beauty is in the discussion that arises so quickly after the answer. And the discussion has just gotten even more beautiful - or at least more organized. Check out Gary Hewitt's latest Numberplay Commenting Fix. Back to the puzzle. Our discussion this week starts off with the following: Heights of a Triangle Can a triangle have heights of lengths 1, 2 and 3? Recap: Sides of a Triangle Last week we encountered the delicious Sides of a Triangle. The puzzle: Can a triangle have sides of lengths 1, 2 and 3? The solution: reader deo got straight to the beauty of the problem. If Numberplay were scored I think the judges would have stopped right there and given deo all possible points. Here's deo: Thinking "out loud," on a plane, no, on a sphere yes. That is, a side of three on a diameter can be connected by sides of one and two at a point above (or below) the diameter. I think. Deo was onto something with the sphere, which Rich in Atlanta and Hans, among others, explored in detail for most of the week. Adam proposed one of the early sphere-based solutions: One configuration that could work (but I don't think could be called a triangle) would be the 3-side lying on the surface of a sphere and the 1- and 2-sides cutting across the interior. The result would look sort of like a lopsided ice-cream cone. But wait. Let's return to deo's plane for a moment. You can't get out a piece of paper and and draw a triangle of sides 1, 2 and 3? Here's Giovanni Ciriani: The answer is no, because the length of the longest side needs to be shorter than the sum of the other two. Ah. What a great way to put it. "Bad triangle, good drawbridge," observed Joe Mahoney. Larry Eisenberg captured the moment with this limerick: With the law of cosines, I see, The angle opposite side 3 Is one eighty degrees The question's a tease, A triangle like that can't be. But what about the flip side of the answer? How could a triangle have sides 1, 2 and 3? Among the many ideas that emerged: Gary: "Yes, but the triangle must be concave." Pradeep Mutalik: "What's the least count of your measuring instrument? Okay, my triangle's sides are 1.00001, 2.00001 and 3 mm. Matt: "sides of 1 inch, 2 centimeters, and 3 (arbitrary unit < 1.787 inches) could be a possible solution." Betsy: "Yes, if the triangle is the musical triangle instrument. One side must be open on a musical triangle by definition. And could anything be more beautiful than this? Here's Dr W via e-mail: Suppose we position three straight line segments of length 1, 2 and 3 in the (x,y,z) Cartesian coordinate system so all ends lie at points on each of the three coordinate axes, as follows: Length 1 from A (0, 0, 1) to B (0, 0,0) Length 2 from B (0, 0, 0) to C (2, 0,0) Length 3 from C (2, 0, 0) to D (0, sqrt (5), 0) Now sight along a line stretched between points A and D. The projection of the three lines along this line of sight forms a closed triangle with non-zero area It's all a matter of perspective. A beautiful perspective, and a fitting way to close the week. This week's Lengths of a Triangle and last week's Sides of a Triangle were warm-up problems offered recently by the Stanford Math Circle. Let's hear it for last week's participants. In order of appearance: Dr W, deo, Adam, Giovani Ciriani, Rich in Atlanta, Pradeep Mutalik, polymath, Golden Dragon, Joshua Zucker, Gary, Jules, Matt, Marco M, Hans, Betsy, Larry Eisenberg, Ravi, Neil M, Joe Mahoney, KWRegan, Stephen Weber, alexandra b, Brian C and Gary A. Here's that link again to Gary Hewitt's Numberplay Commenting Fix. If you'd like to suggest improvements please post as a comment to this post or e-mail Numberplay@NYTimes.com. Like to take a crack at your own Numberplay Commenting Fix? Mr. Hewitt suggests checking out The Times" Developer Network API Tools page and JavaScript InfoVis Toolkit. If you have a favorite puzzle you'd like to share, please e-mail to Numberplay@NYTimes.com.