How Many Squares on a Checkerboard?

So you researched the company you are interviewing for and found out the chances of encountering a non-technical interview question are pretty high. This article is meant to provide a comprehensive answer to the tricky how many squares on a checkerboard question. The short, straight-forward, and simple answer is 64 squares, 32 white ones and 32 black ones. But no interviewer will be satisfied with this answer because this question also refers to the squares formed by combining all the little black and white squares. The answer they are looking for is 204, but let?s see how we got to that number.

How Many Squares on a Checkerboard: Explained

To provide an accurate answer, we need to first approach the question logically. We already know there are 64 little squares on a checkerboard, so now we need to find out how many larger squares there are.

Note: We want to count ALL the squares on a checkerboard; this includes the squares that overlap. ?

Let?s start with the 2×2 squares on a checkerboard. The right vertical border has 7 2×2 squares with their rightmost border on this edge, and the left horizontal border also has 7. If 2×2 squares take up 7 different positions vertically and other 7 positions horizontally, this means we will have 49 2×2 squares.

If we do the same calculations for the other squares, we will find out a checkerboard has 36 3×3 squares, 25 4×4 squares, 16 5×5 squares, 9 6×6 squares, 4 7×7 squares, and 1 big, 8×8 square. Therefore, the total number of squares on a checkerboard in 204.

How Many Squares on an NxN Checkerboard?

There?s a pattern in the numbers above that most math enthusiasts will notice. Specifically, the number of squares is the sum of squares from 1² to 8². therefore, this is the generalized sum for the total number of squares on an NxN checkerboard:

n² + (n-1)² + (n-2)² + ?. (1)²

We can also use the following formula to determine how many squares we have on an NxN checkerboard:

n(n + 1)(2n + 1)/6

Conclusion

When preparing for an IT interview, it?s important to also cultivate adjacent skills and abilities. The many sub-branches of mathematics like algebra, calculus, combinatorics, and the like. This was just one of the many examples of non-technical interview questions you can stumble upon when interviewing for a new job.