The puzzle I shared on my last post was very simple, and although it hinted at the topic that I originally wanted to write about I think I can do better by slightly modifying it.

Since the last post was in Spanish I’ll re-share the full text of the puzzle again, this time in English, and highlight the single modification in red, bold font:

One of the mechanics in the garage has a son in high school who is a very bright student and very good in mathematics and computer programming. He stopped by the garage one day after school and his father asked what he was doing in school and his son told him about his latest assignment.

He is supposed to write a computer program to handle very large numbers that could not be handled on a typical hand-held calculator. The teacher told the students to use that program to determine if a certain very large number is a perfect square. (What is a perfect square? A perfect square is a whole number or an integer that is arrived at by squaring another whole number. For example, 900 is a perfect square of 30; 196 is a perfect square of 14. 625 is the perfect square of 25. So there are no fractions, no decimals, no nothing. Just whole numbers allowed.)

Each student is assigned a particular number. This kid’s number is 334,912,740,121,560. And the teacher wants to know if this is a perfect square.

His father says, “That’s a big number!”

And then out of the inky shadows, who appears but Crusty! And he says, “Oh, your teacher gave you an easy number.”

“She did?” said the kid.

“Oh yeah. I can give you the answer right now.”

The question is, what did Crusty know?

Again, the original puzzle comes from NPR’s Car Talk.

Update #1: Reader Carlos Encarnación writes with an interesting answer: “Crusty knows that 334,912,740,121,560 is divisible by 3, since the sum of all its digits equals 48. On the other hand, he also notes that it isn’t divisible by 9. Therefore 334,912,740,121,560 is not a perfect square. In general, if p is prime and N is divisible by p, but N is not divisible by p^2, then N is not a perfect square. The proof is trivial, it follows immediately from the fundamental theorem of arithmetic and the definition of perfect square.”

Update #2: Readers numerate and Caphi point out that one easy way to check is to notice that the number is divisible by 10 but not by 100 (i.e., it ends with only one zero). This also follows from the answer on Update #1.

Related posts:

• Cuadrado Perfecto (10/March/2011) [in Spanish]
• Vedic Square (17/March/2011)