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Given the following sequence of integer pairs:

$(1,3), (2,4), (3,1), (?,?)$

Which of the following pairs takes the place of $(?,?)$?:

$(-2,4)$
$(-1,4)$
$(-3,2)$
$(-1,3)$
$(-2,3)$

Note that I don't know the answer, and was hoping any of you could shed a light.

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  • $\begingroup$ I've tried to clean up the formatting, but wasn't sure: were the first digits of the options meant to be negative, or were the dashes just pseudo-bullet points? $\endgroup$
    – Alconja
    Nov 1, 2017 at 2:04
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    $\begingroup$ This seems like far too little information to determine a definite correct answer. Unless there's some super-clever thing I'm missing that nails it down, this is going to be a matter of guessing what pattern whoever wrote the question happened to be thinking of. $\endgroup$
    – Gareth McCaughan ♦
    Nov 1, 2017 at 2:05
  • $\begingroup$ I am so sorry, the first digits were meant to be negative, completely slipped my mind, somehow. Corrected the OP $\endgroup$
    – bambozzler3000
    Nov 1, 2017 at 13:11
  • $\begingroup$ For the reasons above I don't think it's possible to give an actual answer to this question, but here for what it's worth is my best guess at what's in the mind of the person who wrote it: each of the two "steps" taken so far changes the parity of both numbers, and the next "step" needs to do likewise; the only one of the given pairs that does this is (-2,4). This is of course a really unsatisfactory answer, not least because it gives us no reason why (-2,4) would be better than say (0,0). $\endgroup$
    – Gareth McCaughan ♦
    Nov 1, 2017 at 15:19
  • $\begingroup$ Solved. @bambozzler3000 $\endgroup$
    – ABcDexter
    Nov 2, 2017 at 11:57

5 Answers 5

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The three given points are part of

an ellipse,
$(1,3)$ being centre, $(2,4)$ and $ (3,1)$ being the ends of minor and major axis respectively.

Ellipse points

The equation of ellipse is :$$5x^2+5y^2-36y-28x+6xy+52=0$$

The only point which lies inside the ellipse is : $(-1,4)$

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The answer is (-2,4) because the three given points form a right-angled triangle and when we tile the plane with copies of that triangle in the "obvious" way (-2,4) is the only one of the given points that's a vertex of one of these triangles:

enter image description here

Or because (-2,4) is the only one of the given answers that continues the pattern of having both coordinates change by an odd integer at each step.

The answer is (-1,4) because it's the only one of the answers that lies in the ellipse extracted from the given points by ABcDexter (in another answer to this question).

The answer is (-3,2) because this is the only one of the given answers that continues the pattern of having one coordinate change by exactly 1 at each step.

The answer is (-1,3) because this is the only one of the given answers that specifies the difference between two of the given points. (The vector from (3,1) to (2,4) is (-1,3).)

The answer is (-2,3) because, er, it's the only one of the given answers for which there's no plausible brief justification.

(In case it isn't obvious: I do not find any of these adequate reasons for picking one of the given answers over all the rest, nor do I expect anything else to be an adequate reason. The first of the ones above, as I mentioned in a comment, seems the least hopelessly useless one and might be the intended answer.)

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  • $\begingroup$ That's a non-trivial answer and an amazing find also. $\endgroup$
    – ABcDexter
    Nov 7, 2017 at 6:48
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    $\begingroup$ The real point of the answer is exactly that it's not amazing! There's so little to go on in the question that we shouldn't be surprised to find lots of comparably-plausible answers. $\endgroup$
    – Gareth McCaughan ♦
    Nov 7, 2017 at 12:07
  • $\begingroup$ Yes, I second that this question is ambiguous. Shouldn't it be VTCed? $\endgroup$
    – ABcDexter
    Nov 7, 2017 at 12:09
  • $\begingroup$ I would VTC except that as a moderator I can't: if I VTC a question then that just closes it immediately. I don't think this one is so obviously bad that I should be closing it unilaterally, especially as no one else seems to be VTCing it. $\endgroup$
    – Gareth McCaughan ♦
    Nov 7, 2017 at 12:15
  • $\begingroup$ Okay, understood. I have raised a flag, maybe others who've answered feel the same. $\endgroup$
    – ABcDexter
    Nov 7, 2017 at 12:17
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Well, based on the given pairs and edited question, I need to provide an updated answer. Earlier answer is

(2, 4) and (1, 3). The reason is - the absolute difference between the pairs of numbers is 2

And an answer for edited question could be:

(-2, 4)

As, the sequence of pairs follows as below:

( a,b) is the given odd numbered pair, then the immediate even pair is calculated as ( b-a, b+a ). With this logic next pair becomes( 1-3, 1+3 ) or (-2, 4).

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  • $\begingroup$ A subtle problem: the problem says, "Which of the following pairs takes the place of…" where 'takes' is a singular verb instead of 'take', the plural verb, so we are most likely looking for one answer. (If you don't understand what I mean by this, consider 'The boy takes cookies from the jar' as opposed to 'The boys take cookies from the jar'.) $\endgroup$
    – boboquack
    Nov 1, 2017 at 4:32
  • $\begingroup$ @boboquack grammatically either pairs take orpair takes is correct, but not a mix resulting of those. $\endgroup$
    – Mea Culpa Nay
    Nov 1, 2017 at 7:33
  • $\begingroup$ Sorry, I don't quite understand what you just said. Could you clarify? $\endgroup$
    – boboquack
    Nov 1, 2017 at 7:35
  • $\begingroup$ @Apep agreed. As which can mean a plural form, I wish to proceed with that ( a possible case of benefit of doubt!?) $\endgroup$
    – Mea Culpa Nay
    Nov 1, 2017 at 8:29
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    $\begingroup$ @MeaCulpaNay Your proposed rule correctly describes the first step given but not the second. $\endgroup$
    – Gareth McCaughan ♦
    Nov 1, 2017 at 15:17
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Answer

$(−2,4)$

Solution

The first digit of each set is incremental. It starts 1, 2, 3 therefore the next digit is likely to be 4.

The second digit of each are all numbers from 1 to 4 but in a seemingly random order. There is 3, 4 and 1. The only remaining number is 2.

So now we have 4 and 2 $(4,2)$. None of these pairs match and they are all negative values. We are only given a choice of negative numbers.

Negative values are opposite to positive values. I'm guessing since we are inverting the position on the number line we can also invert the order of the numbers. Therefore + becomes - and 4,2 becomes 2,4.

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It is a pair:

$(−1,3)$

Reasoning:

Given three pairs forms right triangle. It is the only pair from five to choose that forms right triangle together with previous two pairs.

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