Math

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Sorry, Hey_You, I missed seeing this. You succeeded perfectly in using the "code" tags to get it to display nicely.

However, saying that it is "a pyramid shape" isn't all that helpful, as there are not really enough rows to give any clear indication of how exactly the row length is varying.

Ditto, in a sense, for the content. With essentially just the three states (1, 2 and 3), the overall information content is low.

I'm sure you have a good scheme for generating the rows. I'm fairly sure I could come up with one, as well. As you merely say it "looks best of it's in a pyramid shape", I'll take that shape to be an epiphenomenon. In fact, the form serves to obscure the generator. I'd say that a simple 1-D cellular automaton will satisfy your pattern. I'm too lazy to reverse engineer it, today.

3 1 1 3 1 2 1 1 1 3 1 2 2 1

That's it.

@lua: Now to explain it. It can be confusing ... The very top row just has a "1" in it. The next row has "1 1". This is describing what is in the previous row. The top row has one "1", so the it is written as "1 1". The third row has "2 1" in it. This is describing what the second row has in it. The row has two "1"s, so you write "2 1". Then the forth row describes the third row. The Third row has one "2" and one "1" in it, so you write "1 2 1 1". And so on. Each row describing what the row before it has in it. Kinda not math, but tricky anyways. A great puzzle to play on friends. You can even vary it, by having a 2 in the first row, but I always use 1.

Cheers.

Probability math problems are my favorite, because they often seem that they can be solved solely by commonplace intuition but usually need a little bit more elegant of an explanation to solve.

The Jailer's paradox is one of them; then there's the Gameshow Host;etc..

The Jailer's paradox goes: There's a judge who knows which one of three prisoners (Alex, Bob, and Mike) will be sentenced to death and there's also a jailer who knows which one will die as well. The prisoners also know that two of them will be allowed free and one of them will be sentenced to death. One of the prisoners, Alex, wants to send a letter to his wife, just in case he's the one that will be put to death, so Alex asks the jailer which one of the other prisoners will go free so that he can send his letter with that free prisoner; and then his wife will be able to read his letter.

The jailer pulls Alex aside and says he can't disclose the name of one of the two prisoners who will be freed, because then Alex's probability of dying will increase from 1/3 to 1/2. Is the jailer right? Explain.

Last edited by abstrusemortal on 06 Nov 2007, 11:24 am, edited 1 time in total.

I love the Gameshow Host. I discovered it when I was reading a biography of Paul Erdos. It's one of those problems that you have to read over three or four times before saying "Oh yeah...now I kinda get it!"

1 3 2 1 1 3 1 1 1 2 3 1 1 3 1 1 2 2 1 1
(OK. I cheated. I looked it up. I'd forgotten this series.)

I was looking at equations that express perfect squares recently. I found the following:

for N > 1,

2 2
N + (N - 1) = 2N - 1

as I recall, hope that's correct!

Can anyone solve the following:

3 3
N + (N - 1) = ?

I wrapped your expressions in "code" tags, which makes them use fixed width fonts, otherwise the exponents go walkabout. Viz:

which is the first one with and extra pair of code tags, so you can see the code tags inside!.

Actually, the first should have a minus sign:

plus it is true for all N, and all reals, and all complex numbers...

The second comes out as:

for instance, which isn't half as much fun.

Your original:

which is also pretty unspectacular.

yeah, I kinda screwed up the format and symbols!

The whole point of these equations is to work out the following formula:


X^Y - (X - 1)^Y = ?

Ah... do you really mean to shift the question into other realms....

When couched as "N^K - (N - 1)^K = ?", I'd assume that we were still talking integer values and integer powers, and I'd just answer that there really isn't anything spectacularly interesting going on. Such a formula can be expanded as a finite series, just using the binomial series. It's not very interesting.

But you have switched to "X^Y - (X - 1)^Y = ?", which now implies that you are no longet talking integers, but probably real numbers. If anything, this is now less interesting. Changing the "N" to "X" doesn't make a lot of difference, but making the power non-integral means the series is no longer finite.

One could now ask a few questions about limits and behaviour as X and/or Y tend to particular values. However, offhand I can't think of anything much to say about them.

Conversely, a similar expression, "(1+X/N)^N", as N tends to infinity, is much more interesting.

(1+X/N)^N=e^x. That equation is used in a lot of places.

Interesting thought... in your equation, can you give an idea of what the graph of "X" against "x" might look like? (For various values of N)

So... I got carried away... ignoring the +/- ambiguity for even values of N, we get:

As N tends to infinity, the graph will tend asymptotically to X=x. (And X=-x for N even.)

Are ya referring to analysis? I wouldn't go so far to say it requires a genius-level IQ. Otoh, I would say that math is hard because it's simple. Meaning there is quite a bit of valid work done. It takes a good chunk of time to go though a significant portion of it. Otoh, something like psychology is easy because it's complex, meaning that we can't really say much about it in any comprehensive way, so there isn't a whole to learn about it. I always thought of it as simply being a matter of patience. IMO, math requires a large amount of time to get through compared to most other disciplines... And people tend to be lazy in this regard.

I think it's has more to do with the way they teach it. The way math is taught now is simply teaching how to apply procedures. Calculus has a lot of procedures that if one follows diligently would allow you to pass the course without needing to gain any understanding of why the procedures and formulas are the way they are. But then, why should I really care about reinventing the mathematical wheel once I gleaned enough of a general idea on how something works like a math formula or procedure for instance? It took some math genius years to come up with the proof in the first place. It's the same reason why I use my calculator for large numbers instead of grinding out the numbers in my head. Once you get the general idea of how something works you can use technology to do the rote procedure executions for you, unless you are on some sort of ego trip. Note, I passed advanced calculus so it isn't as if I'm ranting out of envy or bitterness.

That depends on which calculus course you take. Some of them are more rigorous on understanding and will test you by asking proving questions. If you want my opinion of it, I prefer courses that have an equal compromise between computations and proving (the understanding bit). Besides, if you understand it, you have no need to memorize it; you will know it simply by deriving it. Of course, for a test situation, you might not have time to derive the equations again, and might be forced to memorize it. It might be more accurate to say that you might only use calculators for calculations that are difficult to do in your head. Because 4000000 is a huge number, but multiplying it by another huge number with about the same number of zeroes is not really all that difficult, and shouldn't require a calculator, anymore than you need a calculator to know what 1+1 is. I sincerely despise using a calculator because it makes you rely on it too much. You might encounter a question you don't know how to answer because you don't know how to solve it, and no calculator in the world could help you do that. If you rely on a calculator too much, you might start to believe that you couldn't solve the problem because your calculator wasn't good enough.

Case in point? Consider trying to figure out whether or not 3^5555 + 5^3333 is divisible by 7. If you rely on the calculator too much, you'll punch in the numbers and divide it by 7 to see if you get a decimal point. Then your pocket calculator, having only a maximum of 8 digits, doesn't process such large numbers. So you believe you need a better calculator. But if you don't rely on the calculator, you soon realize that 3^6-1 is divisible by 7, and 3 to any power divisible by 6 divided by 7 always results in a remainder of one (this requires use of the mod concept, but once you understand it, it's not usually difficult to apply), and since 3^5555 = (3^6*925)(3^5), the remainder of 3^5555 when divided by 7 is 1*5 = 5. Continuing in this fashion, you can easily figure out what the remainder is for 5^3333, add the two remainders, and see if you get 7. Which proves that you don't need a calculator to do this kind of question. And for the record, no, I did not use a calculator to figure out whether or not 6 divides 5555 or what the number closest to it was. This kind of mental math can be done quite easily in your head, although I'm not saying this to toot my horn; I'm saying this because it is possible. I am not a human calculator. There are people out there even more extraordinary than myself at that. I know a few of them.

Of course, I consistently like to prove my point by taking my math tests (some of which allow pocket calculators) without any calculators, and do some of the longer or bigger calculations by hand, which isn't much anyways, and complete the test before the end of the period, and get a relatively high score on it--proof that even if we are allowed to use a calculator for a test that might be designed for such purposes, you need not have one to ace it.

You might also be interested to know that most professors or teachers won't dock marks if you don't arrive at a completely refined answer. 32*468 is as good an answer as 14976.