Spotting the Mistake Is the Brilliant Part

Added: 2026-05-05

[00:00:00]Hey guys, this looks like a fun one.

[00:00:01]We're given f ofx + y = f ofx + f of y.

[00:00:05]We're given f of 6 and f of 9, and it wants us to find f of 24. This is day five of math week. I found seven internet math challenges, and we're going to try to solve one per day for a week. And so far, we've been on schedule. And now we're going to do this one. And I think there's a mistake. I don't think this one's possible. But if you want to try it on your own, pause it right now cuz I'm going to solve it in three, two, one. So, let's copy this down. And somehow we got to get up to f of 24. So let's try f of 6 + f of 9.

[00:00:35]That's going to be f of 6 + 9. And 6 + 9 is equal to 15. So when we get this, we'll know what f of 15 is. f of 6 is given. That's equal to 9. And f of 9 is given 16. 9 + 16 is equal to 25. And we have f of 15 is equal to 25. Now we can start over. And this time let's do f of 9. So the x's will become 9. And for f of y, let's do f of 15. And that'll end up being equal to f of 9 + 15. And 9 + 15 is equal to 24. That's what we wanted. f of 24. So f of 9 is equal to 16 plus and an f of 15 is equal to 25.

[00:01:18]16 + 25 is equal to 41. And now we have an answer for f of 24. Let's put a box around it. So this is probably what they expected us to do, but this is wrong.

[00:01:27]Let's put a nope. Let me show you why.

[00:01:29]Using everything given here, we can find a different value for f of 24. This time, let's do f of 6 plus f of 6.

[00:01:37]That'll be equal to f of 6 + 6. And that's equal to f of 12. This f of 6 is equal to 9. And same thing for this f of 6, it's equal to 9. 9 + 9 is equal to 18. So we have f of 12 is equal to 18.

[00:01:51]And now we can start over again. f ofx + y is equal to f ofx plus f of y. And this time for both the f ofx and the f of y, let's plug in f of 12. That'll end up being equal to f of 12 + 12, which is f of 24. f of 12 we just found out is equal to 18. And then we're going to add to that this f of 12, which is also going to be 18. 18 + 18 is equal to 36.

[00:02:14]So we have f of 24 is equal to 36. And f of 24 is what we were trying to solve.

[00:02:20]But now we have a contradiction. Using all the stuff we were given, we were able to find f of 24= 41. But we were also able to get f of 24= 36. This input of 24 provides two different outputs.

[00:02:34]That means this is not a function. This whole problem is flawed. I still think it was a lot of fun to go through it. So that's why I wanted to still make the video, but this is not a possible scenario. How exciting. I kind of knew right away it wouldn't be possible because I was pretty sure this was only going to work if these sets of inputs and outputs were proportional. And I think since they weren't proportional, that's why this didn't work out. If you want to learn more about proportional reasoning, I can highly recommend this course on Brilliant called Proportional Reasoning. You can tell they put a lot of thought in each of the lessons. They really use the interactive element wisely. I also think this is a course the younger students would enjoy. It goes through some fun, logical, foundational concepts. No matter what topics you're interested in, whether it's math, science, data analysis, computer science, or programming, Brilliant has tons of courses on it.

[00:03:20]They have world-class teachers from MIT, Harvard, and Stanford. Even if you already know a topic, it's really fun to see how they go about explaining it and also seeing how they utilize the interactive elements on Brilliant's platform. If you'd like to try Brilliant for free for a full 30 days, visit brilliant.org/andandymath.

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