
How many squares can be found in the checkerboard shown in the figure, below?
Answer:204 There is one 8x8 square, the outer border of the figure. There are four 7x7 squares, one aligned with each corner. Another way to count these four squares is to note that each of their upper left corners is aligned with one of the four corners of the upper left small square. There are nine 6x6 squares. To count them, note that each of their upper left corners is aligned with one of the nine corners of the four small squares in the upper left corner of the figure. Similarly, there are 16 5x5 squares, 25 4x4 squares, etc. The number of squares in the figure is the sum of the squares 1+4+9+16+25+36+49+64, which is 204. In case you're curious, the sum of the first n squares is given by the formula (2n³+3n²+n)/6. Formulas for sums of the first n numbers of a given power are always polynomials of degree one higher than the given power. For example, the sum of fourth powers is given by a fifth degree polynomial. See the "related pages" section, below, for references to pages in this website that explain this a bit further. 
See the NEXT puzzle in the Academic Decathlon 2004 Logic Quiz
Finding Coefficients of formula for Sum of Squares  A method to find the sum of n³, or any higher power of n for that matter, as well as any other series that can be exactly fit by a polynomial of finite order.
Method of Successive Differences to find the coefficients of a polynomial f(k), given a few values of f(k) for successive integers, k.
Find the equation of an order n polynomial that passes through n points
The webmaster and author of this Math Help site is Graeme McRae.