## Thursday, December 09, 2010

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

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)

1. Here's my standard reminder... don't post the answer or any outright spoilers before the deadline of Thursday at 3pm ET. If you know the answer, click the link and submit it to NPR, but don't give it away here. Thank you.

2. Blaine, see my post at the end of last week's comments. Your interpretation seems reasonable, but if correct, the instruction to divided the 4x4 square into 16 individual boxes is redundant, since it is already so divided. Is it possible that the "4x4" reference is the error, and the puzzle really does involve dividing a square into 6 individual boxes (i.e., a 3x2 square)?

3. Lorenzo, I'm pretty sure of the question and my answer.

4. Blaine, your were right on both counts.

5. I've added a diagram of how I interpreted the puzzle.

6. Perhaps it would prove useful later on to have a way to describe such triangles as may exist. One way would be to give each vertex a name. In this case, from left to right and top to bottom, we would have:

A B C D E
F G H I J
K L M N O
P Q R S T
U V W X Y

The smallest triangle would be described as type AFG. And so on...

Chuck

7. Let me suggest the following. Everyone start building your lists to reveal after the deadline. Let's use the following convention:
1) Alphabetize all the vertices in a triangle (e.g. AFG, not AGF).
2) Alphabetize the full list.

8. Blaine, I'm thinking your diagram shows a LOT more diagonals that the question specifies.

9. To my eyes, it follows the directions. Draw 16 squares, draw a big diamond hitting the middle of each side of the big square and finally draw an X along the diagonals of the big square. But if you have a different interpretation, feel free to share your diagram.

10. Blaine is correct. The directions call for 6 diagonal lines to be drawn and that is how many he drew. At a glance, there does appear to be many more.

11. They have corrected the web link, and it now says 16 boxes. We have to assume they mean square boxes, or even rectangular boxes, since they don't say. I think Will clearly said 16 this morning, and I take them as square. Using that, I have counted all the triangles. That was so much fun that I then counted all the squares. But I draw the line at counting all the rectangles!

12. I tried this over and over and kept finding more triangles. The obvious challenge is finding triangles made of multiple triangles and not counting any triangle more than once.

After several systematic (and frustrating) approaches, I believe I have found the correct number, though not before many tears.

Blaine, I'm no dummy, but I have absolutely no idea what you're talking about in your opening comments. Does that mean there's a clue or am I a dummy after all??????

I believe the number, if I am correct, is the basis for an ancient from of birth control.

13. Tommy Boy, we seem to have the same answer. And I too am having trouble with Blaine's clues. Regarding your final comment, I assume it does not refer to age.

14. Lorenzo, your assumption is correct (though it would definitely be one for Guinness).

The last comment, like my puzzle post last Wednesday, is also an old joke.

15. After my first count, I think I get Blaine's clues.
Maybe the diagonals in Blaine's diagram would be easier to trace if it were in black and white and not so many colors?

16. Suggestions have been made earlier by Blaine and others that we should try to describe each triangle by designating it with letters representing its vertices. This is an excellent idea. The whole diagram has 25 vertices, so we can use letters A to Y. Reading across from left to right and then top to bottom, the 5 vertices on the top row would be A B C D E, and the 5 vertices on the last row would be U V W X Y.

For example, the single triangle at the top left corner would be designated as ABG. The single triangle at the bottom right corner would be SXY. The composite triangle (on the upper right corner) that circumscribe 16 single triangles would be designated as AEY. The composite triangle that circumscribe 2 single triangles at the bottom left section would be designated as QUW.

It may also be a good idea to use a shortcut designation to describe composite triangles of different sizes. For example, triangle QUW would be described as QUW(2) while triangle AEY would be AEY(16). This would also facilitate the counting. The composite triangles of various sizes are naturally symmetrical, e.g., there are 4 composite triangles, each with 16 singles; and 8 composites with 8 singles.

17. Lorenzo, assuming we do have the same answer, a little research on your part may shed some light on Blaine's clue.

This may be another way for us all to describe our triangles:

Count unique SQUARES first, starting with the smallest (1x1) and noting the number of triangles in each.

Then count the number of unique 2x2 squares and repeat the process, noting any overlaps.

3x3, 4x4, etc. Could be a shorthand way of describing all triangles. Just a thought.

18. I arrived at the same answer several of you have. It’s funny – I thought I had my final answer last night. But this morning I reviewed it one more time before sending it in and found 4 more triangles I’d overlooked before. No mystery now – I’m very confident of my answer and hit the submit button.

Good puzzle and a little different than we’re used to...

Chuck

19. Chuck, The same four may have eluded me at first, too.

I'm wondering about Tommy Boy's clue - Good one! if it's what I think it is.

A partially coded reference which contains a null leads to my answer.

M29 Satan

20. Chuck, I decided to number each small triangle and then code my answers from that. But my number, even eliminating overlaps, seems higher than T's birth control joke. (That is if I get the joke from way back in my hippie, button wearing days)

21. Okay, so I went back and found four more and I think I have an inkling of some related off key joke about flowers and music. But don't worry Blaine, I think the guys will behave and keep it clean.

22. instead of adding the four triangles it sounds like some of you first missed, i had to subtract 4 that overlapped by mistake... trying not to do this without a bunch of diagrams but had to resort to it, the mind is going in circles!

23. Wow, just discovered 8 more I missed. I see now T that your joke is the exact opposite of my hippie days button.

24. RoRo, I would say that you have arrived at the correct answer. Now try to decipher Blaine's clue.

25. It was really helpful to give each point a letter and to alphabetize them. It helped me find the four I was missing. Thanks for that suggestion. I grew up in the 60's but don't remember any numerical reference to "birth control", but I, too, got the opposite of a "hippies" button expression. I guess I can guess?????

26. I have a new total and I don't get Blaine's clues at all....

27. B_D, I believe Chuck's most recent post makes a reference to Blaine's clue.

28. B_D, upon further consideration, I believe you have made a reference to Blaine's clue too.

29. I'm a new puzzler but I used to love tangrams -- drew the picture on a piece of regular old paper as they were describing the diagram, and I did end up with the right picture. Wasn't I feeling smug when I came up with my initial answer! Then I realized when Will had said "most people" won't get it right, he didn't just mean most regular sane people, he meant even most *puzzle* people -- which meant to me that I should take another look. Came up with a more-than-doubled answer (without a plumb bob or chalk line, mind you!)! I can't wait to see if I'm right.

30. This comment has been removed by the author.

31. What are the odds that Blaine's comments refer to probability/statistics as well as geometry/trigonometry?

32. May I defer your query to the creator of James Bond, Mr. Ian Fleming???

33. Tommy Boy, was there a musical clue in your December 5, 3:21 p.m. post?

34. Both RoRo and jutchnbev made reference to the hippie button being the opposite of the correct answer. I also got the same answer; but I didn't look at the two numbers as being opposites, although the symbols were indeed opposites. As for the birth control issue, it seems either "procedure" would work just fine.

35. Dave, there are 3 parts to a musical clue in that post.

36. Dave, I also believe Blaine's opening comments

Chuck - 12/6 9:07 AM

Barnes_Durco - 12/7 10:03 AM

Tommy Boy - 12/7 8:39 PM

All refer to the same musical clue.

37. At the risk of being a comment hog, I'll throw up one more musical clue since it was the answer of a recent NPR Puzzle.

1999

Also, it's Wednesday. We all worked hard on this week's puzzle so I'll go easy on you with the

Midweek Puzzle Break

38. Just when I finally got the connection between Blaine's clue and the song we are all thinking of, Tommy Boy adds to the mystery by referring to Ian Fleming and 1999.

39. TB - my second comment (Dec 07, 10:03) has no intentional clues, honestly.
I just had found a new answer and it doesn't fit with how I was reading Blaine's clues (which I did elaborate upon in my first post 12/5 17:44)

40. I figure if I solved the puzzle two independent ways and got the same answer, that's good for me. Both involve extremely careful visual inspection. I may not be enough of a hippie to understand all the clues, but I think I got one or two of them. (Maybe?)

41. B_D, "Whodunit???" Three clues to the clues for the price of 1.

42. B_D, looked back at 12/5 17:44. Nice one. I hit that number too and was disappointed when it turned out too low. Are we back to Gershwin again?

43. You guys are being too mysterious, but I've cried too many teardrops on this puzzle already, so I'm done.

44. Dave, hasn't that one been beaten to death?

Here's a different musical clue and a seasonal one -- Adam Sandler ?!?

-- Other Ben

45. I can't wait to see the explanations of the various clues.... I didn't have time to count triangles and could not program a computer if my life depended on it... so I will wait.

46. OK, I got 96 triangles, but there's no way I'm going to attempt to describe my calculations by typing.

I guess I'll just say that I focused on the sizes of the triangles, not their placement. Then I tried to figure out every place that specific size could go.

And although I love me some Question Mark and the Mysterians, that one was beaten like a rug. So I went with Adam Sandler's Chanukah song, since it was a greatest hit of 1996.

And Chanukah 2010 is an eight-day festival of lights than ends, as Tom Lehrer would say, "an hour and a half from now."

-- Other Ben

47. This one isn't really that hard. The triangles are all the same shape. Each one has a single right angle. That right angle has to fall on a vertex of the diagram. By symmetry, there are only six that are really different: A, B, C, G, H, M. At each of those you count the triangles, large and small, that have a right angle there. Then you write those numbers on all the other similar vertices and add them up. I get the same answer as Other Ben.

48. I got 96 too. Yes, 96 Tears et al was beaten to death, but I think it was more of an acknowledgement of understanding Blaine's clue.

Ian Fleming was posted as Mr. Ian Fleming, as in Mister Ian. 1999 an obvious reference to Prince who, for a time, also referred to himself as a symbol.

And fianlly,

Q: What's the best position for couples wishing to avoid pregnancy?

A: Back to back

49. William, we used identical strategies. Working with right angles eliminated the possibility of counting the same triangle more than once.

50. Okay, I didn't intentially clue the '?' in my post. Mine was much more obtuse. A semicircle has a measure of Ï€ radians which is the same as 180 degrees. If you search the digits of pi to 180 places, you find '96'.

I've included a diagram in case people want to visually see the 96 triangles. Also, I deliberately posted a colorful diagram so it would be harder for people to bring the diagram into a paint program and start filling in colors to count them that way. (Sorry, Barnes)

51. Ben, sorry that I overdid it on the ? and the Mysterians clue. I took the lazy way out and figured out the answer from the clues that everybody provided and had a case of premature elation when I got the answer.

I have a pretty extensive collection of LPs from the '60s and '70s and I actually have a copy of the ? and the Mysterians album with 96 tears on it.

52. I also used William's method, plus I went through another time and counted individually triangles composed of 1, 2, 4, 8, 9, and 16 component triangle units.

I read some of the "found 4 more triangles" comments as referring to Clinton being elected to 4 more years in 1996.

My "eXtremely Careful Visual Inspection" = XCVI = 96.

Now to get that blasted repetitive organ riff out of my head.

53. Okay... so somehow I got the right answer (but alas no phone call from Will) but the clues were out of my reach... and I misinterpreted a clue ...the birth control one. I was coming up with 96 and I thought "nein to sex" was close enough to 96... oh boy.

54. In case anyone is having trouble visualizing the triangles they missed (like me) crosswordman blog has a beautiful quilt image.

55. Count the Triangles Solution

Or if you were keeping a list:
ABG, ACG, ACK, ACM, ADS, AEM, AEU, AEY, AFG, AGK, AKM, AMU, APS, AUY, BCG, BEQ, CDI, CEI, CEM, CEO, CGH, CGI, CGM, CHI, CIM, CKM, CKO, CKW, CMO, COW, DEI, EIJ, EIO, EMO, EMY, EQT, EUY, FGK, FIU, GHM, GIM, GIQ, GIS, GJY, GKL, GKM, GKQ, GLM, GMQ, GQS, GVY, HIM, IJO, IMN, IMO, IMS, INO, IOS, IQS, IUX, KLQ, KMQ, KMU, KMW, KOW, KPQ, KQU, KUW, LMQ, MNS, MOS, MOW, MOY, MQR, MQS, MQW, MRS, MSW, MUW, MUY, MWY, NOS, OST, OSY, OWY, PQU, QRW, QSW, QUV, QUW, QVW, RSW, STY, SWX, SWY and SXY

The most common ones missed were the size 4 triangles in the middle and the size 9 triangles.

56. No takers on the code?

"M29 Satan" > 29Satan > 29666

It's a Zip Code. Just for fun Google it.

57. I googled 1029 santa and found a real estate deal. Note the MLS#

Search Results
1029 SANTA ANA St, Laguna Beach, CA 92651 | MLS# P716696
Sale Pending: 3 bed, 3 bath, 2000 sq. ft. house
located at 1029 SANTA ANA St, Laguna Beach, CA 92651 on sale for \$849999. MLS# P716696.
www.redfin.com/CA/Laguna-Beach/1029-Santa.../4898868 - Cached - Similar

Seemed too random.

58. Oh! I forgot the most obvious clue I provided. The title of the post was "Triangles Abound"
Triangles = 9 letters
Abound = 6 letters

59. Well, tomorrow's puzzle is already posted. Assuming it's posted correctly, it's considerably easier than last week's puzzle. I solved it in about five minutes, compared to four days for the triangle puzzle (yes, I'm a little slower with the math puzzles than I am with the word puzzles).

60. This is a great puzzle for kids too.

I was wondering if you have a category for kids/family to work on together. Obviously, some puzzles are too much for kids.

Thanks,Chris