Subject: math theory on consecutive squares
Dear Q and Q:
As a teacher at a school called Educere in Houston, I have a ninth-grade student who discovered the following shortcut last year as an eight-grader. I promised him I would find out if it had ever been published under a mathematician's name, and if not, how to go about getting it published under his name, such as the "Ben Rose" rule of consecutive squares.
What he noticed is that given any two consecutive integers (or n and n+1 for any rational number greater than or equal to 2), the difference between their squares was equal to the sum of the two numbers.
i.e., comparing 2 and 3: the difference between 4 and 9 is 5, or 2+3 comparing 4 and 5: the difference between 16 and 25 is 9, or 4+5 etc.
in general, the difference between consecutive squares:
Again, our question is:
Hi Emily and Ben,
This is a nice observation. It was probably known to the ancient Greeks, perhaps before Pythagoras (500 BC). Their observation would have been based on the fact that the sum of consecutive odd numbers is a perfect square:
All the squares can be obtained this way and thus the
difference between consecutive squares is always an odd number, and (as Ben
noticed) any odd number is the sum of consecutive integers.
The fact that Ben was not the first person to make this observation doesn't take anything away from his achievement. He's doing what students and mathematicians are supposed to do!