Question 84757: A designer, attempting to arrange the characters of his artwork in the form of a square grid with equal number of rows and columns, found that 24 characters were left out. When he tried to add one more row and column, he found that he was short of 25 characters. Can you find the number of characters used by the designer?
Answer by rapaljer(4671)   (Show Source):
You can put this solution on YOUR website! I figure that the designer had 600 pieces. Check it out, by figuring that he/she had a square of 24 by 24, with 24 characters left over. That equals 600. Now, adding one unit to each side of the square, you have a 25 by 25 square, which is 625 square units. That leaves an extra 25 spaces left over.
The final answer is 600. As my attempt to explain how I arrived at this, consider the following well-known illustration that consists of a sequence of dots.
Begin with 1^2, and draw 1 dot.
Next, take 2^2, and draw 4 dots in a 2 by 2 square. You can do this by adding 3 dots to the previous dot.
Next, take 3^2, and draw 9 dots in a 3 by 3 square. You can do this by adding 5 dots to the previous square.
Next, take 4^2, and draw 16 dots in a 4 by 4 square. You can do this by adding 7 dots to the previous square.
Continue the pattern, and see if you can figure out the generalization. It turns out that if you had n^2 dots in the square, you must add 2n+1 dots to get the next square which is (n+1)^2 dots.
This 2n+1 dots represents the sum of the number of dots the designer had left over in the smaller rectangle PLUS the number of dots that were open in the larger rectangle. In other words,
Number of characters in the orginal square =  .
Total number of characters =  characters.
I'm sorry this one was hard to explain!!
R^2 at SCC
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