Draw a 4x4 square. Divide it into 16 individual boxes. Next, draw a diagonal line from the middle of each side of the square to the middle of the adjoining side, forming a diamond. And, finally draw a long diagonal line from each corner of the square to the opposite corner, forming an X.

How many triangles can you find in this figure?I was struggling to think of a smart way to solve this puzzle -- okay, I don't care about solving it except that I have to write this blog post, so really I was struggling to think of a smart way to explain the puzzle -- when it hit me: quarter- and half-square triangles. Those are terms that mean something very specific to quilters.

Here's what the diagram should look like:

If you cut a square of fabric in half along the diagonal, you get two half-square triangles -- if you sew those together, you get a single block with two half-square triangles:

There are two of these full-sized blocks in this puzzle. Here's the second one:

If you cut a square of fabric in half along the diagonal, and then cut it again along the other diagonal, you get four quarter-square triangles of fabric. Sewn up, it's called a "bowtie" block and it looks like this:

Here's what makes this tricky: you have to add up all of these half-square and quarter-square triangles for all squares smaller than 4x4 but larger than 1x1 (those get added in at the end, don't worry). Here are those blocks:

That takes care of all the triangles you get from those basic blocks. Our total so far is 24. Now let's look at the diamond shape we're to draw:

This is the square-within-a-square block, and it adds four more triangles to our total: 28.

We were also told to divide the original 4x4 square into 16 squares, but we need to do that in stages. Let's start with the central horizontal and vertical lines that turn our plain square into a four-patch. That doesn't yield any triangles, but if we add the diagonals, we get the famous "pinwheel" block, which gets our total up to 36:

Back to the square-within-a-square block. We can split the middle diamond in half, two ways. This adds four more triangles, so the total is 40. I've used black fabric to indicate triangles that have already been counted.

We can also divide the inner diamond into quarters -- add the four inner triangles and our running total is 44.

If we add the diagonals to the four-patch, we get four bowtie blocks, which gets our total to 60:

But there are two other sets of horizontal and vertical lines to draw. We can add them to our square-within-a-square block, add diagonals, and we get 4 more triangles. (This is called, unimaginatively, a square-within-a-square-within-a-square block.) Total: 64

Back to the diagram we were first told to draw. If you color each of the 16 1x1 squares with their half-square triangles, you get 32 of the smallest possible triangles.

And our grand total is 96.

I'm hoping that's right. If I were to sew these blocks, here's what the resulting quilt might look like:

Gosh, I hope 96 is the right answer! (I'm reassured that Ross and Henry got the same number, albeit without any virtual fabric.)

Time for ...

P I C K A R A N G E

Here are this week's picks for the ranges:

Fewer than 100 -- Mendo Jim

100 - 200 -- Ross

200 - 300 -- Anonymous

300 - 400 -- Jimel

400 - 500 -- Magdalen

500 - 600 -- Dave

600 - 700

700 - 800 -- DAPF

800 - 900

900 - 1,000

1,000 - 1,100

1,100 - 1,200

1,200 - 1,300

1,300 - 1,400

1,400 - 1,500

1,500 - 1,600 -- David

1,600 - 1,700

1,700 - 1,800

1,800 - 1,900

1,900 - 2,000

2,000 - 2,100

2,100 - 2,200

2,200 - 2,300

2,300 - 2,400

2,400 - 2,500

2,500 - 3,000

3,000 - 3,500

3,500 - 4,000

4,000 - 4,500

4,500 - 5,000

More than 5,000

More than 5,000

Fewer than 100 -- Mendo Jim

100 - 200 -- Ross

200 - 300 -- Anonymous

300 - 400 -- Jimel

400 - 500 -- Magdalen

500 - 600 -- Dave

600 - 700

700 - 800 -- DAPF

800 - 900

900 - 1,000

1,000 - 1,100

1,100 - 1,200

1,200 - 1,300

1,300 - 1,400

1,400 - 1,500

1,500 - 1,600 -- David

1,600 - 1,700

1,700 - 1,800

1,800 - 1,900

1,900 - 2,000

2,000 - 2,100

2,100 - 2,200

2,200 - 2,300

2,300 - 2,400

2,400 - 2,500

2,500 - 3,000

3,000 - 3,500

3,500 - 4,000

4,000 - 4,500

4,500 - 5,000

More than 5,000

More than 5,000

*and*it sets a new record
## 3 comments:

96 was also my answer. I am trying to write an explanation of my method that is understandable.

I am quite willing to be patient while you finish sewing my Range prize. Ross and Henry are probably going to have to pitch in to keep up with weekly production.

Nice exposition (if byting a little heavily on my modem).

I'll be interested to see if Dr. S has a higher answer.

A better challenge than usual, but to maintain my usual negativity, Will didn't have to do much.

I'm not sure how Henry arrived at 96, but Ross did it by enumerating all the unique hypotenuses possible in the grid. Someone on Twitter said she used highlighter markers.

No one should expect that quilt coming out of my sewing room any time soon. The virtual quilt is as much as you're going to get.

Mendo Jim -- you know you have to put the kettle on for tea AFTER you start loading the NPR blog posts!

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