An unwritten law of math is that no matter how long you’ve studied it, you can always find a problem that really baffles you, despite it looking simple. You know that there must be an answer but it eludes you. Sometimes, even though you think there's a simple answer, the actual answer is neither simple nor unique.
Professor Bryan Gaensler, an astronomer at the University of Toronto, recently had this experience while trying to help his son with some math homework. He shared the problem on Twitter and many came to his rescue, providing both serious and silly solutions.
The geometry question shows two similar triangles and gives information about two sides of the first triangle and the other side of the second one. The goal is to find the length of the other sides, labeled a, b, and c. The problem seems like it has an obvious solution. It’s just a proportion problem, right? That’s exactly what my first instinct told me. But there is a missing ingredient. The triangle with two sides hasn’t got a known angle, which means it's not unique. So there isn't a single solution to the problem. There is an infinite number of them.
This was pointed out by several people on Twitter. Among them, Chris Solnordal produced a beautiful graphic to explain three possible solutions. This went down particularly well with the people following the thread. Meanwhile, some other suggestions were a lot less serious.
Fellow astrophysicist Professor Peter Coles suggested using a ruler to solve the problem. And to be fair, since the question doesn’t have a “the image is not to scale” label you can’t really disagree with his approach. Jason Marson cleverly highlighted the letters a, b, and c with a bright green “found them”.
Hadi Papei, another astrophysicist, wanted to make sure we're assuming that the universe is flat. Many astro papers begin with that assumption as it seems to be the most likely explanation. Just so you know, a triangle would still need an angle to be unique in curve geometry, but the sum of its internal angles would no longer be 180 degrees.
Math problems with infinite or indefinite solutions never seem to be obvious but they are actually very common. They've even been immortalized in pop culture staples like Mean Girls. “The limit does not exist” should remind us that for any problem, there could be one solution, no solution, or infinite ones.