P.Mean: Counting squares (created 2013-04-02). News: Sign up for "The Monthly Mean," the newsletter that dares to call itself average, www.pmean.com/news.

The Harvard Business Review blog presented an image (see below) and asked you to count the number of squares in the picture, explain how you arrived at that number, and explain the "connection (if any) do you see between this exercise and breakthrough innovation." This was to be done in the comments section of the blog entry. Sometimes I like these exercises and sometimes not, but this one caught my attention, in part because I came up with an answer that I later realized was wrong. I was determined to do this well the second time around. I came up with 30 squares. Here's what I wrote.

1. 30 squares. 2. Well, 16 small (1 by 1) squares, of course, to start. I was nervous about missing some of bigger the squares, but then I realized that every two by two square would have a "+" in the middle. So I counted the number of pluses (9). Then I realized that every 3 by 3 square would have a single square at its center, so I counted the number of single squares not at the edge (4). Then, of course, there is one big 4 by 4 square and 16 little 1 by 1 squares. I got some psychic satisfaction in noting the pattern of numeric squares (16+9+4+1) that gave me more confidence in my answer. 3. As a statistician, I worry a lot about the accuracy of counting, and if there is a way to count differently in a less error prone manner, then I am all for it. Looking for the "+" at the center of each two by two square was a somewhat innovative way to count these squares more accurately. Note the number of people who have a smaller number than 30. They almost always seem to miscount the number of 2 by 2 squares. It is easy to miss a few of these because of all the overlap. The "+' centers do not overlap and are thus easier to count accurately.

If you look at the entries, you see that 30 is probably the most common response, but the problem is somewhat ill-posed. What exactly is a square? If it is an region of a single color with a border of a different color, then the 2 by 2 and larger "squares" don't count anymore. The larger "squares" have a mix of black and white pixels in their interior. Some people see small black squares at the intersection of every vertical and horizontal black line. So how you count depends in large part in what qualifies as a "square."

You can find the blog entry and my response (assuming it gets published), at
--> http://blogs.hbr.org/cs/2013/04/whats_the_connection_between_c.html