Failure Excites Me More Than Success When Doing Mathematics
The last few days, I have been taking a crack at some tough problems that have some kind of clickbait description that “only __% can solve these”. I am not sure if the percentages are true, but these problems aren’t easy.
Before I share with you my surprising reaction to not solving the problem in the picture above, let me share with you a problem of perhaps similar difficulty that I did solve.
(x + 3)(x + 5)(x + 7)(x + 9) = 9
To be honest, I tend to recognize problems like this one because I know what NOT to do — expand the brackets. With multiple brackets having some similarity, the mathematical trick is to make an algebraic substitution that “collapses” the brackets into something easier to work with.
I went with z = x — 6(6 being the average of the constants in the brackets)
(z — 3)(z — 1)(z + 1)(z + 3) = 9
(z — 1)(z + 1)(z — 3)(z + 3) = 9
I won’t give the rest of it away, but the critical first move was done successfully and solving for z with seeing two sets of brackets as difference of squares gives you something you can expand with the constant 9 cancelling out on both sides. Some factoring and simple algebra is all there is left.
And yes, I got a dopamine hit of getting the right answer. However, any “victory” was temporary. It wasn’t really satisfying. I couldn’t explain it.
And then I could.
That’s because the problem in the image was beyond me. I tried some things, but wasn’t getting anywhere. After a while, I decided to watch the solution.
Wow! I loved it! The creativity with the exponents and the clever use of “1”.
I didn’t solve this problem, but I experienced more joy coming out of failure than success. That’s because I learned something NEW!
Even at my age of 61, getting stumped by mathematics is far more of a thrill than solving questions correctly. I don’t want affirmation of things I already understand — I want affirmation of NOT understanding techniques/skills in mathematics.
Here is the solution to the problem.
It’s so elegant with applying the same step over and over until…
Got to love mathematics!
