The Lemon Fool

The quadratic equation

8*x^2 - 8*x + 1 = 0

has two real roots. Given that the 2023rd digit of one of the roots is 7, what is the 2023rd digit of the other root?

Happy new year to all readers.

Cinelli

The solutions are A = sqrt(1/8) + 1/2 and B = sqrt(1/8) - 1/2.

Since these differ by exactly one, the recurring decimal expansions of A + (-B) must add up to 1. But being irrational, they'll never make that neatly, and the per-digit sum is a recurring 0.99999...

So the digit in question is 2, being 9-7.

Now, how the heck did you come up with that 7?

Blame the very simple error in that answer on the New Year celebrations last night!

Other than that, it works.


Julian F. G. W.

I resign.

Specifically, re-sign the formulation of the two roots, making the solution even simpler via an equivalent argument.

I’m glad UncleEbenezer came back with a second post because I have been scratching my head wondering how he managed to get the right answer despite not finding the roots correctly. Actually you don’t need to find the roots (although if you do calculate the roots and look at the first few digits, it gives a strong hint to the solution of this problem) - it is enough to know their sum and that they are both positive. You can find both these pieces of information merely from the coefficients of the quadratic.

But I must give an acknowledgement to the source of this problem. It is from Michael Penn, an American math(s) professor who has released hundreds of maths videos on YouTube. If you have not seen him, usually he takes a problem from a maths competition and solves it on the blackboard. Some of his solutions are outrageous and they involve half a dozen changes of variables to make the evaluation of integrals easier. He has another channel too where he and others teach various topics.

However I must say that in this particular case I think he makes rather a meal of his solution. Why did I choose 7? In Michael’s video, he has 6 in 1994, the year of the competition. I chose 7 randomly – its value wasn’t important. There was a comment on YouTube where someone had listed all 1994 digits of the roots. What a pity it didn’t go up to 2023. Apparently there is software which allows you to find square roots to arbitrary precision. One such is Reduce, a package I once had access to.

But that’s a good place to stop.

Cinelli

cinelli wrote:I’m glad UncleEbenezer came back with a second post because I have been scratching my head wondering how he managed to get the right answer despite not finding the roots correctly.

The incorrect root (in fact a root of (8x^2 + 8x + 1 = 0)) just reverses the sign. A-B becomes B-A. The sign doesn't affect the digits. The result of subtracting it from the correct root based on the difference is no different to adding the two correct roots based on the sum.

Hence my resignation post. Though I shouldn't be posting now, after polishing off that bottle of Primitivo for recycling tomorrow. Talking of which, I must go and put the boxes out.

I chose 7 randomly – its value wasn’t important.

Not important to the form of the solution, but now we know not to trust a wordnumber you say!