Math

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An interesting problem. But somewhat insufficiently stated.

To flesh out the "60 floors" part of the question. It's rather important to state exactly how many distinct distances are involved. It's going to be 60, 61 or 62, depending on whether you include the roof and/or ground floor. Also, if you only counted "rolling the ball off from exactly at floor level", you might only have 59 (no counting the roof and immediately ignoring the ground floor, as a drop from zero height is rather unlikely to cause a ball to break.

The next problem is that you say "the smallest number of trials". If that means minimising the worst case number of trials, forget all the above, and the answer's eleven. (I hope)

No! Stop! I refuse to address this logic problem until the weekend! I have a lot of schoolwork this week, and it will bug me and distract me to no end, my intrigue must wait!

Who cares?

That would be mathematicians.

I started to post the following:
"Math" is the erroneous removal of what is mistakenly though to be a pluralising "s" which is actually an integral part of an abbreviation. If you were to attempt to be consistentent, you would describe yourself as a "U" citizen.

But, I do like to confirm my smart-ass information before I post.

The form "math" is indeed restricted to N. America, but was in use by 1847, as "math.", and had dropped the point to become plain "math" by 1878. The "maths." form only dates from 1911, losing the point in 1917.

Actually, on consideration, I just don't like either abbreviation. I do mathematics.

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Oh, and PS.

If one had a third billiard ball, one would only need at most seven trials.

A fourth ball is no help.

With five, you only need the minimal six trials.

That's interesting though because my professor was saying about how using three billiard balls would be nearly impossible. I'm not quite sure what he meant though.

I got 11 too for the two biliard balls, but out of curiosity, how did you arrive at your answer?

Recurrence relations...

Think of the function F(b,t), which is the number of floors you can do, given "b" balls, in "t" trials.

F(0,t)=0 ... you can't do ANY floors if you haven't got any balls!
F(b,0)=0 ... you can't do ANY floors if you don't do any trials!
F(b,t+1)=1+F(b-1,t)+F(b,t) ... the important relation...

This says that, if you are allowed an extra trial, the number of floors tested goes up according to:
one, for the floor you tested (you now have your result for that floor).
plus, if that doesn't break, you can test a count of the floors above, with a full set of balls, but one less trial remaining.
plus, if it did break, for the floors below, you can only test with one less ball (and trial).

Without going into any more detail, F(b,t) is:

The choice of "60 floors" for the problem is quite neat, as none of the interesting bits care whether you are within one or two of that figure, so whether you count ground floor and/or roof doesn't matter.

As to exactly what the general formula is... it's the t-th degree polynomial whose first t points go through the values 2^j-1. Constant, linear, triangular numbers, then a bit more fiddly.

I really like math and I am from the US. I could go on and on talking about it back in High School. It was my favorite subject, and Physics. They were absolute. Being from the US, I can agree with what is said about the US that Math isn't spoken about that much. At Uni it was in the Math department, but elswhere, it was always about the latest game that just came out or about computers.

Like others have said, I too have found Math to come easily. I was on the Math team in high school. In class I would be following what the teacher is doing and it would be a long huge complex equation and I would raise my hand and point out an error. Everyone in class looked at me. I had a sense that they didn't like me that much or thought me as a freak. Though, I showed them. It was psychology class and we had to take an IQ test. There was this one Math problem so complex and hard that it took me about a minute to do, but on in the class could do it, not of the teacher. I had to explain how it was done.

I have a problem for you all. The problem looks best of it's in a pyramid shape, but it is rather difficult to do. I will try my best.

What is the next row? The first row consiting of "1", the second row consists of "1 1", so the 9th row will have what in it? Good luck.

It's kind of sad that our society is in such as state that mathematics, the cornerstone of science and civilization, is feared and hated. By even myself, at one time, ashamed as it makes me to admit it.

I did the specific case for just two balls and 60 floors. The idea was that you can minimize the trials by having the largest number of trials also be the same as the smallest number of trials; meaning regardless of whatever floor the ball breaks on, you would still have to go through the same number of trials in order to find out which floor it breaks on. And I did this by finding out that if n is the number of trials required, then I would have to try n-1 trials above the nth floor I first tried if the ball didn't break (since the first trial on the nth floor counts as one), and if it didn't break, then I'd try the next n-2, and so on, until you can only go one floor above, which should be the 60th floor. I found a formula for that and found a close match for it, which had n=11, although it might be similar to your idea.

This is neither possible nor a good way to think about the problem.
Consider if there were only two floors to be tested. If you do your first trial on the lower one, and it breaks, there's no need for a second trial. Conversely, if you do your first trial on the higher floor, and it doesn't break, you again need no second trial.
In each case, it the ball does the opposite, you must do a second trial.
I.e. you need to do at most two trials, but half the time, you can get away with one.
In any case, with the 60 floor, two ball problem, if the first floor you try (11)causes the ball to break, you must then test each floor, starting from the bottom (as you've only got the one ball left). However, you may then find that the ball breaks on that bottom floor! Hence you've only done two trials, but have solved the problem.


These are the triangular numbers (think pool/snooker triangle). The sum of the natural numbers un to n, with the result n(n+1)/2.
The next column (three balls) boils down to the expression n(n^2+5)/6.
For four balls, n(n^3-2n^2+11n+14)/24.
I'm not sure what the general formula might be.

I wasn't intending to use that formula for all general cases, since the question only asked us to find out for only 60 floors and two balls. I'll concede this formula might have come up a bit short-sighted, since I wasn't thinking of a more general case.

I often tell people that I'm a mathematician, and that I don't do sums.

Mathematics is ALWAYS about the general cases - the more general the better.

Typically, applying the general theory it to specific cases (like the "60" of this problem) becomes quite trivial and actually easier than picking through the particukar instance on its own.

It also gives you confidence in the answer, because you can verify, easily, that the simple cases - the ones that it is easy to see - come out correctly.

I'd agree with that. Sometimes, people don't understand how it is I can't get a simple question, yet still manage to ace the test. Since I think I have Aspergers, certain aspects of my mathematical thinking are much more highly developed, while other parts of them aren't. I have difficulty keeping tabs on that, unfortunately, but I can understand theories and stuff in math one way, which probably prevents me from understanding the other aspects of the theory because I fail to see it differently.

What people cannot understand, they sometimes distrust. Higher mathematics is simply too taxing for most people; most people do not have the genius-level IQ or higher required to pass Advanced Calculus with even a C. As a result, there is a large chance that mathematics will be looked upon with either suspicion or dislike by a large percentage of the population. (Surveys show that it is.)

You'll hear sometimes in the news that certain schools, such as in California, are watering down math and giving kids points for wrong but 'creative' answers. The reasons cited for their watering down the math they teach are of course not helpful: to prevent the child from having his self-esteem hurt or to raise the average grade in the classroom. These reasons do not lead to preparting the student for the real world. However, I argue that by watering mathematics down to make it easier for the average person attending grade school and college, the math's image is improved. When more people have a better experience with math (and success at solving problems for their level of skill), it will be viewed with less suspicion or dislike. Then, those who can do Advanced Calculus, such as engineers and mathematicians, will be given more funding to do their studies, because their use of mathematics in their research will be considered more of a benefit to society.

I agree with what you say. People do tend to do that, hence, I always just never really socialized. People picked on me, and I just left them alone and did my own thing.

I can't stand the thought of "watering down" maths. I am a logical person, so to me numbers are the law. Numbers are truth/ To say that 2 + 2 = 5 (for example) gets half points is wrong. I saw a special on math awhile back and the reporters did a study on how kids learn it, and they found that there are ways to improve the learning of maths, but the teachers don't do it, so I think it's more teacher teaching it wrongly than anything else. If they taught it right, there would be no need to water it down. *sigh* Some teachers are brilliant, but we also must admit that some are not good. The not so good ones are where we get the expression: Those that can't do, teach. I think we should try to get teachers to teach it better. But how to do that?

I guess no one can figure out my puzzle in the my earlier post in this thread. It's a hard one.

The way I view something that is difficult to understand is that it is probably something like a riddle; it exists so that I can ponder it and begin to understand it; not so that I can turn away from it and distrust it. I don't believe math, whether you're a genius or not, comes easily to everyone. Some of these geniuses simply spent most of their childhood lives reading math textbooks; perhaps the only reason we call them geniuses. I also don't believe in watering down. At first, it might seem appealing to water it down only so that people might be less suspicious towards it or treat it with less distrust, but then people would only be learning a watered down version of math, which means they might be ill prepared for university, or university level math courses get watered down, and there can be little or few advancements made in math, since people's knowledge and experience with math is less advanced and limited.

People used to tell me that calculus would be hard, but I didn't find it that way. I actually enjoyed it.

Since I would be interested in teaching math, I would also like to make math seem more enjoyable or easier to understand when I teach it.

I also used to have a very interesting math 12 teacher, perhaps one of the other reasons why I'm quite interested in math. If anyone knows him, he jokingly calls himself "God" in class.