Open-Ended Math Questions?

[Earthborn]

I don't think there are any simple mathematical problems that are really open-ended. They probably wouldn't really be mathematical.

But there are a whole host of them that for some reason or other entice people to disagree with eachother. Take for example the old "Is 0.999... repeating equal to 1?" (answer: yes), or the Boy/Girl Problem (answer: 1/2, although it is possible to very subtly reformulate the question to get 1/3 and many people tend to read the problem that way).

And then there are those questions that cause problems from other subtleties than language ones, such as these:

Answer:

Here is a similar problem in a picture I once sent Randi:

I have since found out that these types of puzzles are called Fibonacci Jigsaws

 

[Pidge]

The list:
9, 4
6, 6
18, 2
12, 3
1, 36

In case anyone else was interested.

Quibble - Assuming you only use the set of postive integers...

(which is the common convention for stating peoples' ages)

 

[Zombified]

Proof: all numbers are interesting

It depends on if you are limiting your answer to integers, or if you are looking for real numbers as well.

The first (lowest) uninteresting integer is 41, and the first uninteresting real number is 17.41. However, this is in dispute. To some, the fact that the string "41" is included in the numeral "17.41" is in and of itself interesting. To these people, the first uninteresting real number would be 17.92.

All numbers are interesting. You can prove this by induction. Obviously 1 is interesting, 2 is interesting because it's prime and so on. Now, if there are any uninteresting natural numbers, there must be a smallest uninteresting number. But smallest is an interesting property, so that number cannot be uninteresting. This is a contradiction. Therefore, there are no uninteresting natural numbers.

Since all integers are differences between natural numbers, all integers are the difference between two interesting natural numbers. And all rational numbers are the ratios of two interesting integers. Algebraic numbers are the solutions of polynomials with interesting coefficients, and real numbers are the limits of sequences with interesting terms.

Therefore, all numbers are interesting.

 

[ReFLeX]

Was she down 10 compared to where she could have been at the end? yup.

No. You are trying to go around our assumptions that the question was asking the simple arithmetic question. You are making your own assumptions such as the horse having a fixed market value.

I suspect the problem the teacher had was the previous times she had presented it she had used even more ambiguous language than 'financial outcome'.

The question is to be found on many websites labelled "Horse Dealing" and it always uses the words "financial outcome". She herself had the question on a piece of acetate projecting onto a screen so we read it for ourselves. No room for variation there.

Certainly there is only one right answer if you are asking how much more money she has at the end compared to the start.

If no abnormal attention was brought to this question, almost anyone would simply assume that.

AS, the question was not for 8th graders, it was posed to college students and adults.

Because something about it apparently screws up certain types of people's thinking. I would guess she first saw it at some teacher workshop.

 

[pgwenthold]

To be honest, you won't find many "open-ended" maths questions, because if a question has more than one way of being interpreted then it's usually considered a bad question, for being imprecise.

I think any problem involving pi_hat would have to be considered open ended, since it is not known whether pi_hat exists.

(for those who don't know, pi_hat is defined something like, if somewhere in the decimal string for pi, there are 100 "9"s in a row, then pi_hat is the number that you get when you terminate pi after the 100th 9 - since we don't know if there is any part of the decimal string of pi that has 100 9s in a row, we don't know if pi_hat exists (we have not found 100 9s in a row in any of the decimal expansion, but obviously it has not been carried out infinitely)

 

[ReFLeX]

I don't think there are any simple mathematical problems that are really open-ended. They probably wouldn't really be mathematical.

But there are a whole host of them that for some reason or other entice people to disagree with eachother. Take for example the old "Is 0.999... repeating equal to 1?" (answer: yes), or the Boy/Girl Problem (answer: 1/2, although it is possible to very subtly reformulate the question to get 1/3 and many people tend to read the problem that way).

I already mentioned the Boy/Girl problem to her. But I forgot... one of the first things I thought of that day were the old Randi commentary puzzles he used to do. I hadn't looked at any yet. But most of them involved diagrams if I remember right.

 

[Zombified]

I think any problem involving pi_hat would have to be considered open ended, since it is not known whether pi_hat exists.

(for those who don't know, pi_hat is defined something like, if somewhere in the decimal string for pi, there are 100 "9"s in a row, then pi_hat is the number that you get when you terminate pi after the 100th 9 - since we don't know if there is any part of the decimal string of pi that has 100 9s in a row, we don't know if pi_hat exists (we have not found 100 9s in a row in any of the decimal expansion, but obviously it has not been carried out infinitely)

Hmm, I thought you could prove that any finite digit string occurred somewhere in the expansion of pi. I may have to go look that up...

A fun brain-twister is the constant Omega. Basically, it's defined to be the probability that a random bitstream in some encoding of a Turing-complete programming language represents a program that stops. This is a well-defined number that has a definite value, but cannot be computed to more than a certain number of digits, due to Turing's proof that the halting problem is unsolveable (which is equivalent to Godel's theorem). It's an example of a non-computable number - a number for which no algorithm exists to compute it to a given precision. Most familiar transcendental numbers like e and pi are still computable, but the computable numbers are countable while the set of reals is not. There was a post about this on Good Math/Bad Math a few weeks ago.

Edit to add: that's probably well outside the scope of what you're thinking of, but its still interesting.

 

[AmateurScientist]

AS, the question was not for 8th graders, it was posed to college students and adults.

Look at Reflex's post above. Reflex's teacher said it's meant for 10-year-olds.

AS

 

[rjh01]

A question with two possible valid answers is what is the square root of a positive number, eg 4? The valid answers for 4 are 2 and -2. If you go into higher powers then you get even more valid answers.

There is also chaos theory. This includes the "butterfly effect," where a small change in the inputs has a large change in the outputs.

 

[calebprime]

Part of the problem is she gets an extra $10 in the middle step with no indication of where it came from. This complicates things. Essentially we don't really know how much she started with. Let's call it X, which has to be greater than the $50 she initialy spent.

Before the first purchase, she has $X, and 0 horses.

Step 1) she has $X-50, and 1 horse.

Step 2) she has $(X-50)+60==$X+10, and 0 horses

Step 3) she has $(X+10)-70==$X-60 and 1 horse

Step 4) she has $(X-60)+80==$X+20 and 0 horses

So she ends up with $20 more than what she started with.

I got -10.

here's why. Post number #5 shows the reasoning that leads to an end profit of $20. But she could have made $30, since obviously the market price of the horse was $80.

Yes, this all hinges on what you mean by "lost" - no need to post a 'correction' of my reasoning. She was clearly $20 ahead at the end, but she could have been $30 ahead. Profit, or loss? Undefined by the problem statement.

I too was thinking along these lines. It seemed obvious that she was up $20, but what if she got the money from Tony Soprano? What if the horse needed hay? What if she took it the rendering plant?

 

[Mashuna]

Bah. A mathematician has no use for a balance sheet. That's for accountant weenies. The answer is always "0." How exciting.

AS

It's true, the answer's always 0. I'm just not sure why I have such a nightmare actually getting it to that stage.

Still, that's what materiality calculations are for [/dullaccountancybit]

 

[MetalPig]

A question with two possible valid answers is what is the square root of a positive number, eg 4? The valid answers for 4 are 2 and -2.

I don't think so. It's true that -2 squared is 4, but I don't think the square root of 4 can be -2.

[edit]
<EDIT>
Hm, apparently they are both square roots. The positive one is called the principal square root, and the other one isn't.

 

[NobbyNobbs]

I think any problem involving pi_hat would have to be considered open ended, since it is not known whether pi_hat exists.

(for those who don't know, pi_hat is defined something like, if somewhere in the decimal string for pi, there are 100 "9"s in a row, then pi_hat is the number that you get when you terminate pi after the 100th 9 - since we don't know if there is any part of the decimal string of pi that has 100 9s in a row, we don't know if pi_hat exists (we have not found 100 9s in a row in any of the decimal expansion, but obviously it has not been carried out infinitely)

Question: since pi is carried out to an infinite number of places, and doesn't repeat, doesn't pi_hat *have* to exist? We just don't know where.

It's like saying if there are an infinite number of universes, there has to be one where purple monkeys are the dominant lifeform on earth. Or one where George W. Bush is intelligent.

Well, maybe not.

 

[sphenisc]

Question: since pi is carried out to an infinite number of places, and doesn't repeat, doesn't pi_hat *have* to exist? We just don't know where.

It's like saying if there are an infinite number of universes, there has to be one where purple monkeys are the dominant lifeform on earth. Or one where George W. Bush is intelligent.

Well, maybe not.

I don't think so,

If you take the decimal expansion of pi and score out all the 9s, you'd get
3.141526535873.......

This would still have an infinite number of places and not repeat, yet would never be capable of producing the equivalent of pi_hat.

Infinite places and non-repetition are insufficient to prove the presence of 100 9s in a row.

I think.

 

[fribble]

I don't think so,

If you take the decimal expansion of pi and score out all the 9s, you'd get
3.141526535873.......

This would still have an infinite number of places and not repeat, yet would never be capable of producing the equivalent of pi_hat.

Infinite places and non-repetition are insufficient to prove the presence of 100 9s in a row.

I think.

You are correct. I think that proof of pi_hat's existence would have to rely on the notion of how "random" the digits are. For example, if you could prove that given the first n digits of pi, the next digit has an equal chance of being any of 1, 2, ..., 9, 0, then pi_hat's existence follows immediately. (In fact, all you need is to show that 9 has a non-zero probability of occuring.) And I believe this has been proven...

ETA: I take it back. According to Wikipedia, it's not yet known.

 

[drkitten]

Question: since pi is carried out to an infinite number of places, and doesn't repeat, doesn't pi_hat *have* to exist? We just don't know where.

No. The number 0.1010010001000100001000001.....

can also be carried out to an infinite number of places, but no 9 appears anywhere in that expansion.

There's a technical term -- the much overused "normal" -- to describe irrational numbers that do have the kind of random digit expansions so any string will appear somewhere. It's not known (although it's strongly conjectured) that pi is normal.

 

[Soapy Sam]

I don't think the OP contains a mathematics question.

I think it contains an arithmetic question , which is not the same thing at all.

Arithmetic questions all have correct answers, assuming no information is concealed.

Mathematical questions are often actually questions about definitions- the interesting number question is a clear example, as is the 0.999 = 1 question, which hinges on how we define the terms "number" and "infinity". To anyone who accepts the mathematical definitions of those terms , the answer is "yes". To anyone who thinks (as I do) that the term "infinity" is meaningless, the answer is "No".
Math questions are no different from any linguistic questions. The answer depends on what the words mean. You might express the question in mathematical symbols, but that simply adds an extra layer of translation.

 

[roger]

No. You are trying to go around our assumptions that the question was asking the simple arithmetic question. You are making your own assumptions such as the horse having a fixed market value.

It's almost like there is more than one way to interpret the problem .

 

[drkitten]

I don't think the OP contains a mathematics question.

I think it contains an arithmetic question , which is not the same thing at all.

Arithmetic questions all have correct answers, assuming no information is concealed.

Mathematical questions are often actually questions about definitions- the interesting number question is a clear example, as is the 0.999 = 1 question, which hinges on how we define the terms "number" and "infinity". To anyone who accepts the mathematical definitions of those terms , the answer is "yes". To anyone who thinks (as I do) that the term "infinity" is meaningless, the answer is "No".
Math questions are no different from any linguistic questions. The answer depends on what the words mean. You might express the question in mathematical symbols, but that simply adds an extra layer of translation.

This is gibberish. Arithmetic is a form of mathematics; the question in the OP hinges on the understanding of the term "financial outcome." (I've seen identical questions posed using different terms as well.) Similarly, the "average" problem I posed above is an arithmetic problem where the correct answer depends on your understanding of the term "average."

The definitions used in arithmetic are usually sufficiently simple that few people are confused by them -- until you get into questions like "how long is half a piece of chalk," where the meaning of "half a piece" is not clear and needs to be examined.

The only difference -- and it's more apparent than actual -- is that students studying arithmetic are still learning the mechanics of numerical manipulation, and so still need to be tested on it. But the whole point of story problems, from elementary school on up, is to learn to manipulate the linguistic definitions as necessary. What does it reallymean when I say "John gives Mary two apples"? Does that mean that the student should add or subtract?

And the reason that story problems are considered so difficult is because they combine both aspects -- students who are not comfortable with addition alone will hardly be happy with addition plus definitional issues.....

 

[ReFLeX]

It's almost like there is more than one way to interpret the problem .

But that's my complaint! Ambiguity doesn't make for a good open-ended problem, the way I see it. If people are going to debate over an answer, they should at least be responding to the same question!