Making triangles with Six toothpicks

"Use six toothpicks to make four identical triangles."

This is a problem that was given to my pre-algebra math class in Jr. High School. The problem was straight out of an old math book.  It was presented to us by a substitute teacher whose strategy was to offer free time at the end of class if we accomplished certain tasks. The math book listed only one solution to this problem. It stated that the only way to form these triangles was to build them into a tetrahedron (a 3D object). In other words, this old text book presented this problem in order to get kids to think in terms beyond the 2D dimensional realm of a flat piece of paper. 

No one in the class figured out the 3D solution. When the teacher presented the 3D solution, he declared that this was the only solution possible. Several people in the class protested by saying that other solutions had to exist. So, he gave us a double or nothing wager. If someone could figure out a 2D solution to the problem by the next day, he would double the amount of free time he offered.

I worked the problem for a couple of hours that night, not because I wanted the free time, but because it really bugged me. I drew many sketches of possible methods, but none produced four identical triangles while the toothpick ends touched. Then it hit me. Nothing in the problem stated that the toothpicks couldn’t overlap. (Nor did the problem suggest that the triangles had to be equilateral.) I drew up my solution: two toothpicks formed an X, and then the other four toothpicks formed a square around that X with overlapping tips. Easy. I read over the problem several times to make sure my solution was in compliance.

The next day, I left my sketched solution on the teacher’s desk and sat down. A minute or two after class started, the teacher discovered my solution on his desk. He was surprised. He showed our class the solution on the chalkboard. Everyone was happy. Our class won our free time. I didn’t really care. I got my rush the night before by solving the problem.

The next day, a deskmate of mine in another class mentioned the incident. She had the same substitute teacher at a different time in the day, who challenged her class with the same problem. So, the teacher presented my solution to her class as well, using my name. She told me that people in her class were upset with her because she sat next to me in another class and didn’t get the solution from me.

Now note, if you look for this problem these days, there are many different versions online. The problem is stated much more specifically, so as to limit the possible answers. These days, this same problem is worded something like this,

“Using six toothpicks, make four identical equilateral triangles and nothing else. (In other words you can’t make six equilateral triangles, or four triangles and a diamond, etc.)”. (backup link)

Of course, being this specific, the only answer is a tetrahedron. Another similar problem I found allows for several 2D solutions (backup link), but of course it also requires equilateral triangles. However, the solutions are similar to my solution to the old problem. 

Now, I’m not saying I was the first person to figure out this solution.  However, I did figure it out on my own in one night; in the days before the Internet. Looking over the Internet these days, I can’t even find this problem improperly stated. Maybe the writers of that old math book just didn’t do their research, tried to dumb down the problem too much, or just didn’t catch the wording error? I can’t imagine that this problem was improperly stated for hundreds of years before it found its way onto my desk.

Bourne to miss

Dang it. I had plans with Miriam after work on Friday to go to see the Bourne movie that's still in the theaters. Well, she wanted to go get something to eat at a Mexican rest'rant she really likes near her place. The drinks there where stronger than expected. Neither of us were sober, so driving anywhere was out of the question. Dang it! Well, it was still fun. She wanted to look for toys for her new hamster, so we walked to the per supplies store next store.
Then it got crazy. We found these rubber chickens (dressed in beach wear) that made this awful noise wailing when squeezed. It was hilarious. We made a symphony of wailing rubber chickens. We got a photo video and audio recording of it on her phone, but can't figure out how to get it onto the computer yet. When we do, I'll post here.
After that, we called some friends that could pick us up and take us to the movie. Ronnie and Fern ended up coming, but headed to the wrong city at first. Anyways, by the time they got there, it was too late to go to even the evening showing, so we decided just to head back to her place. Allie came by and brought the Knocked-up DVD to watch. It was kinda funny. It ended around 12am, and we all pretty much headed home right after.
I want to see Bourne in the theater. Oh well.

Great void in the universe

So, here's the deal. Science has disputed the age of the universe for a long time now. The current consensus is that it's about 12 Billion years so or, +/- a billion years (depending upon methodology). This comes from the previous observations that when looking into deep space, the farthest objects are about 12 billion light years away, in every direction.
When I was much younger, this interpretation of observations of the Universe got me thinking. How do we know the Universe is only as old as the number of light years away we can see? The only way to equate seeing the farthest objects with the age of the Universe is if we can also see one or both of the following: the center of the Universe; the edge of the Universe. Here's the problem. No one has claimed to observe either. Without this frame of reference, there is no way to determine the Universe's size. Without knowing how big the Universe is, we can't really know its age, unless by some magical fluke, we are at the center of the Universe. OK, so a general idea of the age of the Universe can be determined by the objects we observe at it's visible edge (Quasars), but we don't know if this visible edge is really the actual extent of the Universe. At most, we only know that galaxies of some sort began forming at that point in time, 12 billion years ago.
What does this hafta do with the vast void recently discovered in the Universe? The void is estimated as 1 billion light years across. That's 24th the size of the known Universe. That's like looking at a two foot long rulers and spacing them 1 inch apart by length. What would cause this big of a hole in the Universe? Could it be the middle of the Universe. Is it the void left as matter accelerates outward? Or is it the actual edge of the Universe? The notions being put forth by astrophysicist right now is some sort of dark energy explanation. But why does it have to be so exotic? OK, so I don't think it's the edge of the Universe because I believe astronomers do see deep objects past this void. So, hey, why couldn't it be the center of the Universe left void from matter expanding outward? One way to test this is to examine redshifts of the objects between us and this void to see if there are subtly less than objects in every other direction, using Hubble's Law. If the void is the center of the Universe, the age of the Universe could be judged with extreme accuracy because we'd no longer have to relay on objects 12 billion light years away. We'd only need to observe our distance from the center, then calculate the mass based on the shape of the Universe and the fact that the Universe looks the same in every direction.
This is all just an idea. I don't know much more than the press story. Heck, for all I know, this void could be where god lives. lol Just some fun to call up my old thoughts about the Universe.

Koi Pool

On Saturday, my new brother in-law planned a small suprize for his mother. My wife took their mom to The City to shop. Afterwards, her dad, brother, he's best friend and I met them for a big dinner at Koi Palace. Allie had only told her that she was talking her to a noodle place for dinner, so it ended up being a pretty good surprize.

On Sunday, Allie and I went over to Miriam's place and chilled by the pool for awhile, then had great chicken dinner and watched a movie.

Today, it looks like we are going to head up to The City just to be out and about on this Labor Day.

Linkin Minutes

After listening to the Linkin Park's Minutes to Midnight release, I am a bit disappointed. I agree with reviews that state they've gone from being original and fresh in a genre they pretty much created into trying to cover a style that has been done by hundreds of other bands. I miss the rock-rap combination they were so talented at mixing. I welcome their evolution and change, but that usually means exploring new areas and not covering stuff that's been done over and over. In addition, their effect in the tired 90's still alternative rock isn't a good contribution. They added nothing to the genre, and just copied what others have already done (and not well at that). So, I'm disappointed. I feel the CD is still enjoyable, but it's just not up to the standard fans have come to expect from LP.

By contrast, when Green Day came back, they also changed. They also went with a political message. But the changes they made improved their style and showcased their talents in music writing and playing. Green Day did the transition well. Linkin Park failed. Before this point, I would've put both group on each footing in becoming great bands. If Linkin Park can recover from Minutes to Midnight, them more power to them. At this moment, it doesn't appear they will be a band with longevity.

Lunar Eclipse

I saw the lunar eclipse this morning. I didn't plan on it, but thought it would be cool to see it. By a strange coincidence I woke up this morning pretty much at the peek of the eclipse at 3:40am. Got dressed and went outside. Spooky. I expected to see something like a new moon silhouetted against the night sky. What I saw was pure evi...., well, no it wasn't evil at all. It was cool though. It was like looking at the full moon, but with the lights turned out, as though the man on the moon forget to switch the light bulb on. It was a dark brown color. A new moon is generally very dim and flat in appearance. The eclipsed moon has the full effect of a full moon, just a lot dimmer and dirtier looking. I shared the site with Allie from our bathroom window a few minutes later, since she woke up too. This full moon eclipse was more spectacular than most to astronomers because it was so long. Unfortunately, that means it was all the more boring to watch for the rest of us. I got my glimpse and went back to bed. Can check "See lunar eclipse." off my to-do list.