# yay math!



## cmhardw (Apr 18, 2008)

This has been discovered before, but I independently "discovered" (more accurately I guess I "verified") the formula given on the following page today after about 3 hours of working with it.

http://mathworld.wolfram.com/TriangleTiling.html

The formula counts the number of triangles, including inverted ones, in a stack of triangles all arranged into one larger triangle in a regular pattern (see the above link).

At the tutoring center where I work with we often give younger kids this problem with 4 rows and ask them how many triangles there are. The answer we are looking for is 27, but most kids give the answer of 16 (counting only the little triangles). Sometimes kids guess 17 (counting the large triangle itself). However, once you show them one of the groups of triangles inside the shape most get the correct number of 27 without any other help.

I had my final formula written in the more condensed form:
floor[(k)(k+2)(2K+1)/8]

I verified that the quantity k(k+2)(2k+1) is evenly divisible by 8 when k is even and is congruent to 1 (mod 8) when k is odd. This justifies the use of the floor function as not reducing it to the lower integer for odd numbers.

I also proved the independent even and odd formulas via mathematical induction, the same form the formulas are in on the mathworld page. I know this is not new material, but I sure had fun "discovering" it!

I just think it's funny how a question on a children's worksheet fascinated me to the point to spend 3 hours figuring it out in the general case ;-)

Anyway, I'm glad I can now quickly figure out the answer to how many triangles are in larger sized triangle grids like these for the problem we give. Now I can appear to be a genius when showing the students about the larger sized triangle grids haha ;-) Not to mention if I ever take an IQ test and this question is on it, correct answer in like 3 seconds! ;-)

And for the record I don't believe in intelligence being a trait given by birth, but sought after by choice.

Chris


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## Hadley4000 (Apr 18, 2008)

That is crazy.

I REALLY hate math.


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## badmephisto (Apr 18, 2008)

interesting problem. I try to stay away from Number Theory when possible 
i couldnt figure out the formula on spot, but im confident that I could write a recursive algorithm that finds the solution


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## Johannes91 (Apr 18, 2008)

Project Euler's problem #163 is a bit similar. Haven't thought much about it yet.


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## cmhardw (Apr 18, 2008)

badmephisto said:


> interesting problem. I try to stay away from Number Theory when possible
> i couldnt figure out the formula on spot, but im confident that I could write a recursive algorithm that finds the solution



If you do want to try this problem, which I recommend since it was a lot of fun, try to find a recursive function for finding a(n+2) knowing a(n). Although I've written that in function notation I mean the (n+2) and the (n) to be subscripts. Do this for even and odd triangles. That's how I got to the formula (after a couple of dead ends before that).

Chris


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## Swordsman Kirby (Apr 18, 2008)

Ugh I just did a problem similar to this three days ago...

Good thing I came out to the same answer.


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## Dene (Apr 18, 2008)

Sounds like fun! But I hate mathematical induction >.< . I try to stick with calculus.


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## pcharles93 (Apr 19, 2008)

Wow, that's pretty cool. I thought my ability to multiply numbers in the 75-100, 100-115, and 115-130 groups was impressive. Now that I typed that and reading it now, I realize how useless my life has been so far.


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## Leo (Apr 19, 2008)

I think math can be pretty cool sometimes, but unfortunately with the way the school system is..I'm a year behind in math. I was on a block schedule at my old school, moved here halfway through my freshman year, where classes are year long rather than semester (I would've taken Algebra my second semester at my old school) so here they were already 1/2 through the course, and thought I wouldn't have been able to keep up, so the shoved me in to pre algebra . Now im in algebra as a sophomore and will have to take math as a senior (most seniors don't need it any more by then here).

But on topic, I miss those things where you had to count all of the shapes in them


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## cmhardw (Apr 19, 2008)

pcharles93 said:


> Wow, that's pretty cool. I thought my ability to multiply numbers in the 75-100, 100-115, and 115-130 groups was impressive. Now that I typed that and reading it now, I realize how useless my life has been so far.



Actually my point was not to brag but to show that I think it's interesting how this kind of question shows up on IQ tests, and yet once you have solved it to find the formula you sort of defeat the purpose of what IQ tests are looking for.

This is one reason why I think the idea of IQ is overrated, because someone could be trained from an early age to solve IQ test kinds of problems. Then, once given an IQ test, they would have an IQ over 200 or something, when maybe their general ability in all areas is not as strong as that test score would predict.

I meant to place a strong emphasis on mentioning that knowing this formula will only let me *appear* to be a genius. I don't claim myself to be one, but using this formula I could defeat this problem in seconds on an IQ test with very little mental effort. Before studying this problem I would potentially have gotten that question wrong, or have taken a long time to figure it out and wasted time that I could have spent on other problems.

The problem itself I find interesting, but also the idea of studying IQ test kinds of problems in order to boost your score. For the record I have no interested in actually taking an IQ test. However, I think it's funny how now I could solve such a question effortlessly, when before studying this problem I might potentially have gotten it wrong on a timed IQ test.

Chris


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## Dene (Apr 20, 2008)

IQ tests, like any other distribution, are designed to look at a population, not an individual! You should know that Mr. Hardwick!


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## alexc (Apr 20, 2008)

Nice job, I guess... I hate math, I have no idea what that formula even means or what you guys are even talking about....


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## cmhardw (Apr 20, 2008)

Dene said:


> IQ tests, like any other distribution, are designed to look at a population, not an individual! You should know that Mr. Hardwick!



Yes true, it is a measure of how you have done on the test when compared to all others who have taken the same test. But my question is this: is the purpose of the IQ test itself to see how well you can solve problems presented to you which you have *never seen before, or how well you can solve problems that you may or may not have studied before?

I have always assumed that the spirit of an IQ test was to give the tester as many problems as possible that will be presented in as unfamiliar a format as possible. Sure they may mimic problems seen in real life, but they are always presented in a way where the information is encoded into an abstract picture, or something like that.

Again that is just my opinion, but to me memorizing a formula and using that formula to correcetly solve a problem on an IQ test is artificially boosting my score in relation to how I would have potentially done had that type of problem been presented to me in a way that I found unfamiliar. Now take that example to the extreme of a child raised since they were very young to solve exactly the kinds of question seen on the IQ test, such that the test itself contained no unfamiliar questions at all. Does the score on this person's IQ test accurately reflect their intelligence level? That was really the basis for why I brought the IQ test part into this.

Chris*


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## Mike Hughey (Apr 21, 2008)

That's very interesting. When I was 6 years old and starting first grade, they gave me an IQ test, and apparently I was off the charts. (They wound up skipping me up to second grade, and there began a number of years where I was the littlest kid and picked on like crazy, but that's another story. ) I'm pretty sure the reason I did so well on the IQ test was that I had already learned to read, and I had checked out several books from the library with typical brain teasers and logic puzzles, because I thought it was fun to solve them. So the problems on the IQ test I was given were almost entirely familiar and very similar to problems I had already solved on my own at home, sometimes looking at solutions to see how they were done. So my off-the-charts IQ score was probably COMPLETELY unfair - it was like I had studied for it! My more recent "IQ scores" (SATs, etc.) have been much closer to normal, which confirms this idea.

I would imagine that the majority of cubers are likely to love (and practice) logic puzzles enough that most any IQ test is probably a little unfair, based on what you're saying, Chris.


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## pcharles93 (Apr 21, 2008)

Same with me, Mike. I learned to read and speak English when I was around 4 or 5. Coupled with my knowledge of Vietnamese, I could manipulate and play with words and spent less time on assignments in future classes. This allowed me more time to investigate logic puzzles and math puzzles. In all my free time, I have discovered and proven several of my mathematical discoveries. Nothing big, just some faster ways to multiply big numbers, find squares with addition, and some area formulas. I also have had tons of test taking experience and my faster reading pace helps too. This shocks my teachers around exam time. Besides all of that, I like taking IQ tests in school and surprising my friends. Most IQ tests consist of questions about reading carefully and being able to manipulate words with a few random questions thrown in. Right now, my friends are trying to hide me from Mensa.


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## Dene (Apr 22, 2008)

Hmm, well I think with IQ tests, for it to be done properly, it needs to be done individually with a trained Psychologist who can give you a variety of questions to try to ensure that you aren't familiar with at least some of them.
Of course, you can study for an IQ test, as the more you know the better you will do (and that makes sense, right?). However recieveing a question that you already know the answer to isn't really a test of your "on-the-spot" problem solving abilities. I'm not sure what they do to try and get around this.


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## immortalcube (Apr 26, 2008)

One aspect that you guys might be overlooking here is if intelligence can be improved upon, or if it's entirely dependent on genetics, heredity, etc. Can you get 'smarter'? If that's the case, one could make an argument that doing math puzzles, logic games, etc. is improving one's intelligence, thus the higher scores on the tests. "The brain is like a muscle: the more you exercise it, the stronger it gets..." Assuming that you can train your brain to be smarter, doing logic puzzles and such doesn't mess with the results at all, and even if all my reasoning is crap and completely wrong, doing this type of math stuff is what we've been encouraged to do all along, right? True genius is usually found in the ones that work hard at it, not the ones that 'get' everything right away...


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## cmhardw (Apr 26, 2008)

My personal opinion is that hereditary intelligence is measured by the relative (to the rest of the population) speed and ease at which someone can assimilate information from short term memory into long term memory.

Note that this does not say anything about how much an individual knows, just at how quickly/easily they could "learn" something new.
 
Notice that someone who knows a LOT of information, but is slow to learn is considered someone who is only average intelligence. Someone who knows very little, but is capable of turning short term memory information into long term memory information is considered very intelligent.

I think our current societal view of intelligence is biased toward the quantity of information known, not necessarily toward the speed/ease of learning part. Of course an IQ test measures both, but I feel the test can be "beaten" if you know how to study it correctly.

This is of course only my opinion, having worked with teaching lots of kids math. I have seen some kids who most parents would describe as "not too bright" can work incredibly hard and surpass some of those that most every parent would describe as "very bright" in skill level and knowledge of a subject. I have also seen cases of students who learn incredibly quickly when they choose to, but who are also incredibly lazy and usually choose not to study or learn.

To sum up what I am saying, nearly everyone is capable of reaching an extraordinarily high level of what most people would call "intelligence", but some people can do that more quickly than others. Also, at the very top levels there will be people who learn so much so quickly that someone who cannot convert short term memory information into long term memory information at a comparable level might not possibly ever reach quite that level, but could come close.

Enough of my babble, just my $0.02

Chris


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## Dene (Apr 26, 2008)

cmhardw said:


> Enough of my babble, just my $0.02



Well this is my $4 billion! (I don't really have anything to say, mainly I'm just mocking you, the concept of the whole "$0.02" thing doesn't make any _cents_ to me (lol, awesome pun)).

I should say something of relevance though..... Hmmm....

In my opinion, intelligence only matters if you use it, which is why I have every intention of remaining in the academic community. Discuss.


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