This Geometry Puzzle Has Nine Solutions (Here Are

Here's what catches me about this problem: it looks like the kind of thing you'd breeze through in high school geometry, and then it stops you cold.

The setup is almost aggressively simple. Triangle ABC. Angle C equals 20 degrees. Angle B equals 80 degrees. Point D sits somewhere along BC such that the length CD equals the length AB. Draw line AD. What's angle ADB?

That's it. No exotic shapes, no weird constraints. Just a triangle and a question.

Presh Talwalkar, who runs the YouTube channel MindYourDecisions, recently walked through two methods for solving it. But here's the thing that made me actually pause the video: according to his sources, there are nine documented solutions to this problem. Nine different paths to the same 30-degree answer.

That multiplicity tells you something. Not about the answer—the answer is just 30 degrees—but about the problem itself.

The Geometric Solution (Or: Build Something That Wasn't There)

Talwalkar's first approach does something I find deeply satisfying: it solves the problem by constructing geometry that didn't exist in the original question.

First, the straightforward stuff. Since angles B and C are 80 and 20 degrees respectively, angle A must be 80 degrees (because triangles sum to 180). Two 80-degree angles means we're looking at an isosceles triangle, so BC equals AC.

Then comes what Talwalkar calls "the extraordinary trick."

He constructs an equilateral triangle—point E—using AC as the base. Just builds it right there outside the original triangle. In an equilateral triangle, all angles equal 60 degrees, which means you can start doing angle arithmetic: if angle ECA is 60 degrees and angle BCA is 20 degrees, then angle BCE is 40 degrees. If angle BAC is 80 degrees and angle EAC is 60 degrees, then angle BAE is 20 degrees.

Now you've got triangle BAE, where BC equals CE (because BC equals AC, and AC equals CE). Another isosceles triangle. The vertex angle is 40 degrees, so the base angles each measure 70 degrees.

"Triangle BAE is congruent to triangle DCA by side-angle-side," Talwalkar explains, "because we have the sides AB is equal to CD. The two angles are equal to 20 degrees and the other two sides AE and CA are congruent."

Once you've established that congruence, angle CDA must equal 150 degrees (80 + 70). And since angle ADB is supplementary to that, it's 30 degrees.

The solution is elegant. It's also completely non-obvious. You have to decide to build an equilateral triangle outside your original diagram. That's not deduction—that's invention.

The Trigonometric Sledgehammer

Talwalkar's second method is what he cheerfully calls "overpowered for these geometry problems." You apply the law of sines twice, set up an equation, and solve for x.

Let x equal angle BDA. Using the exterior angle theorem, angle DAC equals x minus 20 degrees. Apply law of sines to triangle ABD: AD/AB equals sin(80)/sin(x). Apply it again to triangle ADC: AD/CD equals sin(20)/sin(x-20). But CD equals AB (given), so both expressions equal AD/AB, which means they equal each other.

Now you've got: sin(20)/sin(x-20) = sin(80)/sin(x).

You could solve this numerically and get x = 30 degrees immediately. But Talwalkar works through the algebra by hand, using the identity sin(80) = cos(10) and the double angle formula to eventually show that both sides reduce to expressions involving sin(30), which equals 1/2.

Then he does something interesting: he proves 30 degrees is the only solution by showing the relevant function is strictly decreasing across the valid range. No other angles work.

"We're not always able to come up with creative solutions," he notes. "And that's why we can just hack it with trigonometry."

There's a tension there that I think is worth sitting with. The geometric solution feels like insight—you see the structure and exploit it. The trigonometric solution feels like machinery—you apply tools until the answer emerges. Both are mathematics. Both are valid. They produce identical results through completely different reasoning.

What Nine Solutions Actually Means

Talwalkar references a 1993 Russian paper by C. Knop that documented nine different approaches to this exact problem. The Cut-the-Knot website maintains an index of these solutions, each using different constructions, different theorems, different ways of seeing the same configuration of lines and angles.

This isn't unusual in mathematics, but it's illustrative. Some problems have a natural solution—one obvious path that most people find. Other problems resist that singularity. They sit at intersections of multiple mathematical domains, accessible through geometry or trigonometry or coordinate systems or vector methods.

The existence of nine solutions suggests this is a problem that mathematicians keep returning to, not because the answer is uncertain, but because the routes keep revealing something new. Each solution is a different argument about what matters in this configuration. The equilateral triangle construction says the key is symmetry and congruence. The trigonometric approach says the key is ratios and equations. They're both right. They're describing different aspects of the same reality.

I think about this in terms of how we teach problem-solving. We often present a problem, show a solution, and move on—as if finding one path closes the question. But problems like this one suggest something else: that the real insight isn't the answer, but the multiplicity of approaches. That mathematics isn't about finding the way, but about understanding why multiple ways work.

"This problem is much harder than it looks," Talwalkar says at the start. He's right, but maybe not for the reason he means. It's not hard because the answer is elusive—it's hard because it doesn't tell you how to see it. You have to bring your own lens, whether that's geometric intuition or algebraic machinery or some other framework entirely.

And once you've solved it one way, the question becomes: can you solve it differently? Can you find the path you didn't take? That's where the real problem lives—not in the 30 degrees, but in the nine ways of getting there.

Nadia Marchetti writes about science, unexplained phenomena, and the questions that resist simple answers.