Fun little math problem

[Gerson W. Barbosa]

(05-11-2016 07:57 AM)Paul Dale Wrote:  

(05-11-2016 06:31 AM)Dieter Wrote:  How did you do it?

I figured the missing values were integers & it was then clear that they must be small. There were not a lot of options left: 2 & 4, 3 & 9, 4 & 16. I got it right first go after this.

It looks like you and Dieter have tackled the problem the same way it was done more than one hundred years ago by young Ramanujan:

"One afternoon back in 1902, during recess, an older student, said to be the smartest in his class, handed him a math problem. Ramanujan was so smart? Well, then, let him solve this: At first glance falling under the familiar heading of “two simultaneous equations in two unknowns,” the problem actually confronted Ramanujan with a difficult fourth-degree equation and meant recalling a theorem applicable to a particular class of them. To any ordinarily smart fourteen-year-old, it would be exceedingly difficult. “To my astonishment,” Rajagopalachari remembered later, “Ramanujan worked it out in half a minute and arrived at the answer by two steps.” In fact, he probably didn’t “work it out” at all, but simply looked at it, guessed the answer might be one where each was a square, tried a couple of possibilities in his head, and saw the solution, x = 9 and y = 4, jump out at him; in other words, it was a piece of fancy footwork, nothing mathematically profound." From "The Man Who Knew Infinity: A Life of the Genius Ramanujan (English Edition)" by Robert Kanigel.