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# Blaine's Puzzle Blog

## Weekly discussion on the NPR puzzler, brain teasers, math problems and more.

Updated: 2018-04-20T09:28:25.080-07:00

NPR Sunday Puzzle (Apr 14, 2013): 90° Letter Rotation

2013-05-04T18:44:53.751-07:00

NPR Sunday Puzzle (Apr 14, 2013): 90° Letter Rotation:
(image) Q: Take a common English word. Write it in capital letters. Move the first letter to the end and rotate it 90 degrees. You'll get a new word that is pronounced exactly the same as the first word. What words are these?
I think it is a foregone conclusion that Will intends us to get creative with how we write our letters.

Edit: The two hints were "foregone" and "write" which contain hints to two possible pairs. The picture gives an example of how you might write a W so it looks like an E when rotated. By the way, the picture is of a set of Ambigrammic Letter Tiles created by Eric Harshbarger.
A: WON, ONE or WRY, RYE

NPR Sunday Puzzle (Oct 7, 2012): Hexagon Diagonals - Count the Triangles

2012-10-11T11:59:30.415-07:00

(image) NPR Sunday Puzzle (Oct 7, 2012): Hexagon Diagonals - Count the Triangles:

Q: Draw a regular hexagon, and connect every pair of vertices except one. The pair you don't connect are not on opposite sides of the hexagon, but along a shorter diagonal. How many triangles of any size are in this figure?
The diagram in the upper right should help. I've removed one diagonal. It looks like a cool cube, don't you think?

Edit: The words "cool cube" were a double hint. First, the diagram I drew reminded me of the isometric cubes in the Q*Bert video game which was released in 1982. Additionally, if you cube the answer (82^3) you get 551368. I think the result is cool because 55+13=68. As a final clue, in several of my comments, I used the word "lead" which happens to be 82 on the Periodic Table of Elements.
A: 82 Triangles - be sure to watch the video for an explanation of the answer.

GeekDad Puzzle of the Week: Waffle Cuts

2018-01-23T01:54:55.521-08:00

GeekDad Puzzle of the Week: Waffle Cuts: (image)
Q: If we only cut along the ridges of a circular waffle, and if each cut traverses the waffle in a straight line from edge to edge, how many different ways can the waffle be cut?
Note: rotations, horizontal flips, and vertical flips of a set of cuts should only be counted once.
Given that there are 6 places to cut vertically and 6 places to cut horizontally, that's a total of 12 cut lines. If you allow for any combination of these 12 lines to be cut or not, you have a total of 2^12 = 4096 ways to divide the waffle. But of course, the puzzle asks for the number of unique ways to cut the waffle, not including any mirrored or rotationally symmetric sets of cuts.

After the official answer to the puzzle is posted, I'll post my solution here.

Edit: The solution is posted, but just the number without any detail. Also, I disagree with their counting of the "no cuts at all" solution as one of the ways to "cut" the waffle. In any case, a full detailing of my solution along with an enumeration of all 665 (or 666) ways to uniquely cut the waffle can be found in Blaine's Solution to the GeekDad Waffle Puzzle.

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

2018-01-23T01:46:11.822-08:00

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).

(image)

• 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

2018-01-23T01:43:34.331-08:00

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

2018-01-22T23:42:51.910-08:00

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 purple circle, which has the bigger area, and why?

Playing with Blocks

2018-01-22T23:09:12.056-08:00

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?