how old is he?

[nanase]

Hi fellas,

I was given this puzzle in school. Here goes :
Jack's great-grandfather was a mathematician.
He was x years old on 26 june in the year x².
How old was Jack's great-grandfather when he published his book, cosmic topology,
on 26 June in the year 1973?


my guess is x =44 and the answer to the question is 37
is it correct? I was doing trial and error using calc.
what is the smart / algebraic way of solving it?

 

[lev888]

Plug in your number, see if it works. Then try 43 and 45.
I don't see a smarter approach.

 

[Dr.Peterson]

I suppose your method was to take the square root of 2019 and round down, in order to find the last year that was a perfect square. That, plus lev888's next step, seems to be about the most mathematical way to do it. I would not call it trial and error; the "guess" is just the largest possible answer, and the "trial" is just checking, with only two or three possibilities to check.

On the other hand, the answer to the question can't possibly be 37; how could he be 44 in 1936 and only 37 in 1973?? Or did you leave out a part of the problem that mentioned time travel?

 

[JeffM]

[MATH]b \text { is year of birth and thus a positive integer.}[/MATH]
[MATH]x \text { is a non-negative integer such that } x = x^2 - b \implies x = \dfrac{1 + \sqrt{1 + 4b}}{2}.[/MATH]
But b is constrained. x is an integer. So the square root of 1 + 4b must be an odd integer. Furthermore b must be less than 1973 by at least a few years and almost certainly more than 1860 by at least a few years. Assume therefore

[MATH]1860 \le b \le 1970.[/MATH]
[MATH]\therefore 7441 \le 1 + 4b \le 7881 \implies 86 < \sqrt{1 + 4b} < 89 \implies \sqrt{1 + 4b} = 87 \implies[/MATH]
[MATH]4b = 87^2 - 1 = 7568 \implies b = 1892.[/MATH]
That would make him 81 when the book was published.

EDIT: By the way, this is just a different way to get the exact same answer as everyone else got.

[MATH]1892 + 44 = 1936 = 44^2.[/MATH]
The only difference is that there is nothing like trial and error involved.

FURTHER EDIT: Denis would have solved this with a three line looper program.

 

[nanase]

Thanks for the various approach, I'm getting different ideas and enriched!
Thanks folks I appreciate it !!