Two Aging Mathematicians
Math brain teasers require computations to solve.
Two aging mathematics professors, Al Gebra and Cal Culus were sitting at their favourite Thai restaurant, Phun Kshuns, reminiscing about when they first met, while their waiter, Theo Rumsanrools (who also took Professor Gebra's Advanced Mathematics course at Cart Eshian University) eavesdropped on their conversation as he brought them their meals.
"You know what occurred to me the other day, Cal?" asked Professor Gebra.
"What's that?" said Professor Culus.
"Well, when we met, the square of your age contained the same three digits as the square of my age, just in a different order," Professor Gebra continued.
"This will really blow your socks off then, Al! If you take the square of the sum of our ages when we met and split it into two 2digit numbers, you'd have my age then and your age now!" Professor Culus exclaimed.
Wanting to impress his professor, Theo interjected.
"I thought you were much younger than that, Professor Gebra," Theo said.
Said Professor Gebra, "And just how old am I then, young fellow?"
Answer
"Well, you're 81, sir," Theo replied.
"Right you are! And how did you figure that out so quickly?" Professor Gebra asked.
"You said the square of your ages when you and Professor Culus met had only three digits, so the oldest you could have been was 31. That means that there were only a few possible ages the two of you could have been. Those being (with the associated squares):
14 & 31 (196 & 961),
13 & 31 (169 & 961),
16 & 25 (256 & 625),
12 & 21 (144 & 441),
13 & 14 (169 & 196).
Now the square of the sum of each of these pair of ages are respectively:
2025,
1936,
1681,
1089,
729.
And from those figures I concluded that Professor Culus must have been 16 when you met and hence you, Professor Gebra, are now 81!" Theo explained.
"Aren't you a bright young fellow! You should come and study my Advanced Mathematics course at Cart Eshian University!" Professor Gebra exclaimed.
"Uh, I already do," Theo said sheepishly.
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Comments
Codammanus
Apr 10, 2003
 You wanna bust my brain, give me a MATH teaser. Yet, with this one, I at least had fun trying! Good 1 Snaps. 
jimrcook
Apr 28, 2003
 Since the splitting into 2, two digit numbers didn't specify the new numbers had to come from the first pair and the second pair, it's possible to use 1936 as the square of (13+31) and 1x3x as the 13 and x9x6 as 96, making the fellow fifteen years older. Just a thought Thanks for the puzzle.

catsclutter
Oct 23, 2003
 you've got a typo in the second line 
puzzpal
Dec 30, 2003
 
puzzpal
Dec 30, 2003
 It was challenging
and entertaining. 
paul726
Mar 02, 2006
 An excellent teaser, even now, three years later. 
lelrod
Jul 26, 2016
 Very good puzzle. Similar to my all time favorite. I solved it in about 5 minutes. 
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