On Facebook, some radio station posted a "squares puzzle" that is simply logic (not even really "math" or "geometry" at all), and they have 10,000 comments arguing about how many squares are in the diagram. The radio station must get some kind of advertising money bonus for getting that many comments. People keep forwarding it to me as a "puzzle." I am about to SCREEEEAAAAM.
I would like to do the "coloring" thing and post a bunch of layers that would have each square colored in separately, so that people could click on each layer as a "card" and see the forty squares, but I don't have the time to spend on that if I am to finish polishing LOVE IS THE BRIDGE and work a bit on the firebug-ghost story *and* keep track of Hubby and his brother on the road trip home. So I'm just going to do a bit of logic and hope that anyone who's still confused by this puzzle and still intrigued can follow it.
There are a few insights that you need to have when you are looking at this. It's like that figure that first looks like a vase, but then you have a shift in your visual paradigm and you can see two faces in profile. Sort of.
First, there's a large square that encloses the 16 small squares. That's 17.
There's an overlay of two four-square-sized squares. Take those off and count them as ten. (If you mentally remove those, you'll have an easier time seeing what I am about to point out.)
There are nine 2x2 squares. I know that only six of those are apparent, but don't forget the ones in the "middle" that also count. This would be really cool done as "layers" in Photoshop that you could click through. Psychedelic!
Now for the insight that sometimes comes late: see the 3x3 squares? One has its left-hand corner in the left-hand corner of the large diagram, and another has its right-hand corner in the RH corner of the large diagram, and another has its lower RH corner in the lower RH of the large diagram, and the last one has its lower LH corner in the lower LH of the large diagram. Here's where those colored layers would REALLY rock and be psychedelic if we cycled through all of it to a cool sitar track. But, again, I don't have the time and resources, so we leave that as an exercise to the very bored reader with time on her hands. LOL
Don't count RECTANGLES, only squares. That would be a different (and more geometrically interesting) puzzle.
SO! After you take off the two overlaid squares (which count as 10--four squares inside each overlaid larger square), then you have 30 remaining. Add the 10 back in and the total is 40.
There is a formula that calculates the number of squares enclosed in a figure; you could probably Google it or Ask Dr. Math. (That Ask Dr. Math website used to belong to Swarthmore and I knew the guy who started it . . . now they've given it to another university where I don't know anyone, but I have a couple of Q/A thingies posted there. Or they were still there last I checked. That is a great site for really advanced questions.)
So, FORTY squares.
Some people felt that it would have to be an odd number because "everything's symmetrical around the horizontal or vertical axis, and there's a match for each of those, and then you count the big one and it makes it odd." But those people missed the 2x2s in the "middle" and the 3x3s, mostly. So it does NOT have to be odd. Logical fallacy in action.
Logic!! It's what's for dinner.