Blaine's Puzzle Blog

NPR Sunday Puzzle (Nov 6, 2011): Count the Equilateral Triangles

2011-11-10T12:00:00.000-08:00

NPR Sunday Puzzle (Nov 6, 2011): Count the Equilateral Triangles:

Q: Take 15 coins. Arrange them in an equilateral triangle with one coin at the top, two coins touching below, three coins below that, then four, then five. Remove the three coins at the corners so you're left with 12 coins. Using the centers of the 12 coins as points, how many equilateral triangles can you find by joining points with lines?

Minnesota is the land of 10,000 lakes, but I know the answer is much smaller than that.

Edit: My hint points to a shorter form of Minnesota, namely the abbreviation MN. That's also the abbreviation for Manganese (Mn) which has an atomic number of 25.

A: 25 equilateral triangles total (see the video for details).

13 small triangles pointing up or down
4 medium triangles pointing up or down
6 medium triangles pointing left or right
2 large triangles at a slight angle

NPR Sunday Puzzle (Dec 5, 2010): Triangles Abound

2010-12-09T13:29:00.000-08:00

NPR Sunday Puzzle (Dec 5, 2010): Triangles Abound:

Q: From Sam Loyd, a puzzle-maker from a century ago: 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?

Getting the answer is really easy; the key is to think of geometry. Let's see, if you start with a square and cut it along the diagonal, you get a triangle. Similarly, if you take a circle and cut a chord through the center, you get a semicircle. Take the measure in radians extended by the measure in degrees and you should have the answer, assuming you haven't made an error. Well, at least that is how I got my answer.

P.S. The NPR website currently has a couple typos in their posted puzzle (e.g. It should be Sam Loyd not Sam Lloyd. And a 4x4 square forms 16 smaller squares instead of 6. I'm pretty sure I have the intended question but be prepared for changes if the on-air puzzle is stated differently.

P.P.S. I've added a diagram now that I've confirmed the wording of the on-air puzzle.

Edit: Okay, I deliberately added a few "faux" clues to my original post in case some people undercounted. A couple common undercounts were 84 and 88. For 84, the misleading hint was "error. Well" hinting at 1984. For 88, there were a couple hints to "key" and "chord" that should make one think of a piano. But the real answer is 96 which was hinted to by this clue: "...get a semicircle. Take the measure in radians (which is pi) extended by the measure in degrees (which is 180°) and you should have the answer..." Now if you take pi and write out the digits 3.141592653589793238... you'll find '96' starting at position 180. You can confirm this by typing '96' into the Pi Search page

A: 96 triangles as enumurated in the following Count the Triangles Solution (PDF)

Hitting the Target Puzzle

2008-04-04T17:02:00.000-07:00

Here's a quick puzzle. In the attached image, a circle is inscribed in a square which is inscribed in another circle.

Of the outside yellow ring, or the inside magenta circle, which has the bigger area, and why?

Playing with Blocks

2008-03-14T13:59:00.000-07:00

Here's a fun puzzle to ponder.

A certain number of faces of a large wooden cube are stained. Then the block is divided into equal-sized smaller cubes. Counting we find that there are exactly 45 smaller cubes that are unstained. How many faces of the big cube were originally stained?

Feel free to add a comment with your answer, along with how you solved it.

Create a foldable 3-D dodecahedron calendar

2008-02-08T17:19:00.000-08:00

I discovered a fascinating site a few years back and completely forgot about it. You've probably seen a 3-D calendar with each month on one of the faces of a dodecahedron. But have you ever wanted to print and construct your own? Ole Arntzen of Norway created a webpage that lets you pick a year, a language and a few other options and then it creates a printable template for a 12-sided calendar.

You can generate a ready-to-print PDF file, or an editable PostScript file. With a little editing of the PS file, you can add holidays, birthdays, school breaks, anniversaries, etc. Take a look!

The Disguised Triangles Revisited...

2007-01-10T21:03:00.000-08:00

A couple months back I posted this picture and asked people to count how many different triangles are in the hexagonal pattern to the right. It is now time to reveal the answer. Click the picture for the solution.

The Disguised Triangles

2006-11-09T17:48:00.000-08:00

How many different triangles can you find in the hexagonal pattern to the right? I believe I correctly counted them all and will post my answer sometime in the future...