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Math and Logic

Why So Symmetrical?

A circle is symmetrical infinitely many ways, but how many lines of symmetry can a square have?

It turns out, there's only four.

Let's consider a diagonal line of symmetry and see how it works. The square below with circles in it is symmetrical along the axis shown, but not along any others.

Imagine folding on the dotted line—everything on both "halves" would match up perfectly. Then, it is easy to see that no other fold can accomplish the same thing. With any other fold, either the circles or edges of the square will fail to align.

What if we wanted to add more symmetry to the figure? Let's give a new, horizontal line of symmetry a try.

Suppose we want to add symmetry across the line shown in yellow above. We would need to add a copy of the circles: You can see them in yellow, a perfect match for the circles below the horizontal, yellow line. Unfortunately, we've created a problem for ourselves: by adding this line of symmetry (and the corresponding circles), we've destroyed the original diagonal line of symmetry. Our new circles wouldn't match up with anything if we folded on the diagonal line. We'll need to add some more circles if we want both of these lines of symmetry.

There! Now we've done it. But alas, these new circles have messed up the horizontal symmetry. Let's try again.

Finally! Now we have two lines of… wait a second. We actually have more than two lines of symmetry now, don't we?  

Maybe we did more work than we needed to? Or maybe this figure was an exception? What do you think? What's the least amount of work we could do that still gives two working lines of symmetry?

Today's Challenge

What is the minimum number of additional unit squares that have to be shaded yellow in the above figure, in order for there to be (at least) two lines of symmetry?

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