(x-x_1)*(x-x_2)*(x-x_3)........(x-x_n) : GMAT Data Sufficiency (DS)
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# (x-x_1)*(x-x_2)*(x-x_3)........(x-x_n)

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04 Dec 2012, 00:11
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This is actually a riddle from some book I had while in the 8th or 9th grade.. I've just tried my best to make it into a DS question... So here goes.. My second question on this forum... No official answer as this is my own question...

If $$0<n\leq{26}$$ and n and x are integers, what is the value of $$(x-x_1)*(x-x_2)*(x-x_3)........(x-x_n)$$?

1)$$x_1 = a, x_2 = b, x_3 = c$$and so on such that $$x_a = a^{th}$$ alphabet when the english alphabet is arranged in alphabetic order and each alphabet represents a unique number.

2) n = 26

[Reveal] Spoiler:

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04 Dec 2012, 11:32
MacFauz wrote:
This is actually a riddle from some book I had while in the 8th or 9th grade.. I've just tried my best to make it into a DS question... So here goes.. My second question on this forum... No official answer as this is my own question...

If $$0<n\leq{26}$$ and n and x are integers, what is the value of $$(x-x_1)*(x-x_2)*(x-x_3)........(x-x_n)$$?

1)$$x_1 = a, x_2 = b, x_3 = c$$and so on such that $$x_a = a^{th}$$ alphabet when the english alphabet is arranged in alphabetic order and each alphabet represents a unique number.

2) n = 26

[Reveal] Spoiler:

Dear MacFauz,
I don't think this question works as is.

The prompt asks "what is the value" of the expression --- this question typically denotes a numerical value. So, we want a number answer. As it stands, we don't know the numerical value of x, and we don't know the numerical value of any of the alphabetical variables.

Statement #1 tells us the letters we will choose for the variables, but that's not the same as getting values for the variables. We have a bunch of variables, and we know the letter will assign to these variables, but we have no idea what the values will be. Any one of them can be any integer.

Statement #2 just tells us there are 26 variables, so 26 factors in the expression.

Even if we knew that the nth variable equaled n, so that the expression were (x-1)*(x-2)*(x-3)* ....*(x-26), we still wouldn't know the value of x itself, and therefore still would not know the value of the expression. It seems to me the answer should be (E) --- nothing is even close to being determined.

Perhaps I am missing something, or misinterpreting something. I would be intrigued to find out how you are thinking about this question.

Mike
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Mike McGarry
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04 Dec 2012, 20:21
mikemcgarry wrote:
MacFauz wrote:
This is actually a riddle from some book I had while in the 8th or 9th grade.. I've just tried my best to make it into a DS question... So here goes.. My second question on this forum... No official answer as this is my own question...

If $$0<n\leq{26}$$ and n and x are integers, what is the value of $$(x-x_1)*(x-x_2)*(x-x_3)........(x-x_n)$$?

1)$$x_1 = a, x_2 = b, x_3 = c$$and so on such that $$x_a = a^{th}$$ alphabet when the english alphabet is arranged in alphabetic order and each alphabet represents a unique number.

2) n = 26

[Reveal] Spoiler:

Dear MacFauz,
I don't think this question works as is.

The prompt asks "what is the value" of the expression --- this question typically denotes a numerical value. So, we want a number answer. As it stands, we don't know the numerical value of x, and we don't know the numerical value of any of the alphabetical variables.

Statement #1 tells us the letters we will choose for the variables, but that's not the same as getting values for the variables. We have a bunch of variables, and we know the letter will assign to these variables, but we have no idea what the values will be. Any one of them can be any integer.

Statement #2 just tells us there are 26 variables, so 26 factors in the expression.

Even if we knew that the nth variable equaled n, so that the expression were (x-1)*(x-2)*(x-3)* ....*(x-26), we still wouldn't know the value of x itself, and therefore still would not know the value of the expression. It seems to me the answer should be (E) --- nothing is even close to being determined.

Perhaps I am missing something, or misinterpreting something. I would be intrigued to find out how you are thinking about this question.

Mike

Thanks for replying Mike... I should say I was worried whether Statement 1 would be able to convey the meaning properly. The riddle went like this : what is the value of (x-a)*(x-b)-(x-c).....(x-z). The answer was 0. The reason being that this series contains the term (x-x). The logic I used to frame the question was:

Statement 1) The series given becomes (x-a)*(x-b)*(x-c).... etc. But we do not know until which letter this series extends. If it ends before x, we cannot determine the value otherwise the value is zero. Insufficient.

Statement 2) The series given becomes $$(x-x_1)*(x-x_2)....(x-x_{26})$$. Insufficient.

1 & 2 together the series is (x-a)*(x-b)*(x-c).......(x-w)*(x-x)*(x-y)*(x-z) = 0.

I understand statement 1 is a bit convoluted. Is there a better way for me to express the meaning clearly or is the question itself not salvageable??
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05 Dec 2012, 10:29
MacFauz wrote:
Thanks for replying Mike... I should say I was worried whether Statement 1 would be able to convey the meaning properly. The riddle went like this : what is the value of (x-a)*(x-b)-(x-c).....(x-z). The answer was 0. The reason being that this series contains the term (x-x). The logic I used to frame the question was:

Statement 1) The series given becomes (x-a)*(x-b)*(x-c).... etc. But we do not know until which letter this series extends. If it ends before x, we cannot determine the value otherwise the value is zero. Insufficient.
Statement 2) The series given becomes $$(x-x_1)*(x-x_2)....(x-x_{26})$$. Insufficient.
1 & 2 together the series is (x-a)*(x-b)*(x-c).......(x-w)*(x-x)*(x-y)*(x-z) = 0.
I understand statement 1 is a bit convoluted. Is there a better way for me to express the meaning clearly or is the question itself not salvageable??

Dear MacFauz
OK, I see what you mean. As odd as this may sound, a good riddle or puzzle typically is not a valid basis for a good GMAT question. I know that may sound paradoxical, since so many GMAT math questions can be puzzling, and many are designed to exploit common misunderstanding. I'm not sure I will be able to explain this distinction.
I guess what makes a good GMAT math problem --- it relies on properties that a mathematician would recognize as important, although it may be framed in a way so as to induce folks to fall into a predictable mathematical trap. By contrast, a riddle or puzzle will often rely on emphasizing the importance of something that others would typically overlook. In this particular problem, the "trick" consisted in the names assigned to the variables. From a purely mathematical point of view, what names we assign to variables is entirely arbitrary, and doesn't affect the underlying mathematics at all. In fact, this is ultimately one of the defining characteristics of mathematical thinking, one hardest for non-mathematicians to appreciate: the absolute arbitrariness of external notation, the absolute fungibility of one variable/symbol for another. Therefore, I imagine that most people skilled with mathematics would entirely overlook the significance of the variable names, as I did. A question that "fools" people who understand mathematics well --- that may well be the characteristic of a good puzzle or riddle, but doesn't not constitute a good GMAT math question. The mark of a good GMAT math question is that it finely discriminates between those who understand math well and those who think about math superficially. In a way, this puzzle-question rewards a kind of superficial thinking (attachment to the names of the variables), which is the opposite of what a good GMAT question does.
I want to say: I really appreciate your effort in putting a question out there. I would say this question is not an appropriate basis for a GMAT question, but don't give up. It's actually excellent practice to understand how GMAT questions are constructed. Let me know if you create any more questions.
Mike
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05 Dec 2012, 20:31
mikemcgarry wrote:
MacFauz wrote:
Thanks for replying Mike... I should say I was worried whether Statement 1 would be able to convey the meaning properly. The riddle went like this : what is the value of (x-a)*(x-b)-(x-c).....(x-z). The answer was 0. The reason being that this series contains the term (x-x). The logic I used to frame the question was:

Statement 1) The series given becomes (x-a)*(x-b)*(x-c).... etc. But we do not know until which letter this series extends. If it ends before x, we cannot determine the value otherwise the value is zero. Insufficient.
Statement 2) The series given becomes $$(x-x_1)*(x-x_2)....(x-x_{26})$$. Insufficient.
1 & 2 together the series is (x-a)*(x-b)*(x-c).......(x-w)*(x-x)*(x-y)*(x-z) = 0.
I understand statement 1 is a bit convoluted. Is there a better way for me to express the meaning clearly or is the question itself not salvageable??

Dear MacFauz
OK, I see what you mean. As odd as this may sound, a good riddle or puzzle typically is not a valid basis for a good GMAT question. I know that may sound paradoxical, since so many GMAT math questions can be puzzling, and many are designed to exploit common misunderstanding. I'm not sure I will be able to explain this distinction.
I guess what makes a good GMAT math problem --- it relies on properties that a mathematician would recognize as important, although it may be framed in a way so as to induce folks to fall into a predictable mathematical trap. By contrast, a riddle or puzzle will often rely on emphasizing the importance of something that others would typically overlook. In this particular problem, the "trick" consisted in the names assigned to the variables. From a purely mathematical point of view, what names we assign to variables is entirely arbitrary, and doesn't affect the underlying mathematics at all. In fact, this is ultimately one of the defining characteristics of mathematical thinking, one hardest for non-mathematicians to appreciate: the absolute arbitrariness of external notation, the absolute fungibility of one variable/symbol for another. Therefore, I imagine that most people skilled with mathematics would entirely overlook the significance of the variable names, as I did. A question that "fools" people who understand mathematics well --- that may well be the characteristic of a good puzzle or riddle, but doesn't not constitute a good GMAT math question. The mark of a good GMAT math question is that it finely discriminates between those who understand math well and those who think about math superficially. In a way, this puzzle-question rewards a kind of superficial thinking (attachment to the names of the variables), which is the opposite of what a good GMAT question does.
I want to say: I really appreciate your effort in putting a question out there. I would say this question is not an appropriate basis for a GMAT question, but don't give up. It's actually excellent practice to understand how GMAT questions are constructed. Let me know if you create any more questions.
Mike

Thanks Mike... And will do for sure...
Just curious though... Is there any way to reframe this question to make it GMAT worthy??
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06 Dec 2012, 00:19
MacFauz wrote:
Just curious though... Is there any way to reframe this question to make it GMAT worthy??

I would say no. It seems the whole question depends so specifically on the choice of letters for variables, something very superficial from a mathematical perspective.

Here's something that occurs to me, two questions that would be GMAT worthy, on the hard side for the GMAT, but they're changed so much from the original that I don't know if you would consider them anything like a reframing:

What is the value of x^8 + 7x^6 - 3x^4 + 14x^2 - 5?
Statement #1: x^2 = 4
Statement #2: x^3 = 8

What is the value of x^7 + 7x^5 - 3x^3 + 14x?
Statement #1: x^2 = 4
Statement #2: x^3 = 8

Mike
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06 Dec 2012, 00:52
Understood..
And my answers to these questions are D and B..

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Re: (x-x_1)*(x-x_2)*(x-x_3)........(x-x_n)   [#permalink] 06 Dec 2012, 00:52
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