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This puzzle is a one I met in an IQ test. I believe it's so tricky, however, the concept is quite simple.

How many minutes is it before 12 noon if 68 minutes ago it was three times as many minutes past 10 a.m.?

suppose x is the minutes before 12 noon.

Now I have 3 meanings with 3 different solutions

  1. … if 68 minutes ago it (then) was three times as many minutes past 10 am (now)

    x + 68 = 3 $\times$ (120 - x) $\rightarrow$ x = 73 min

  2. … if 68 minutes ago it (now) was three times as many minutes past 10 am (then)

    x = 3 $\times$ [120 - (x + 68)] $\rightarrow$ x = 39 min

  3. … if 68 minutes ago it (then) was three times as many minutes past 10 am (then)

    x + 68 = 3 $\times$ [120 - (x + 68)] $\rightarrow$ x = 22 min

But the model answer I was told was 13 which I see definitely wrong. So, my question is what's wrong with my understanding of this ambiguous puzzle?


edit...

My interpretations are backed up on this usage of "times as many"

p.s Please, state how much you agree with potential interpretations?


edit #2...

Source of confusion reduces to how to interpret the phrase: x is three times as y

My interpretation was: x is (equals) three times as ($\times$ 3) y $\Rightarrow$ x = 3 $\times$ y

The opposite interpretation was: x is three times ($\times$ 3) as (equals) y $\Rightarrow$ x $\times$ 3 = y

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    $\begingroup$ xkcd.com/169 $\endgroup$ – Wen1now Jul 3 '17 at 2:36
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    $\begingroup$ Most people seem to be having no issue with interpreting this. Is this perhaps a language barrier issue? maybe somebody could translate this into your first language to help it make more sense. $\endgroup$ – PeterL Jul 3 '17 at 20:05
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    $\begingroup$ @Samha': I agree with the interpretation in that link, and my answer agrees with it. Your answer 2 (the closest to being correct) has the opposite interpretation from your link. Perhaps something else in the wording is confusing you? (It's not to do with being mathematical or logical: I have many years experience of that, and the answer 13 is the mathematical/logical one; if you're getting other answers the problem lies somewhere else :-) ) $\endgroup$ – psmears Jul 3 '17 at 21:11
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    $\begingroup$ For what it's worth, I honestly can't figure out how you get any of the three equations you list (and also got 13 before looking at any written answer here or your proposed answer). $\endgroup$ – Daniel Wagner Jul 4 '17 at 2:14
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    $\begingroup$ @Samha' Aha, thanks for the clarification! Now the problem is clear: "it was" is a very idiomatic shortening of "the time was", and so idiomatic that it completely overwhelms any possible insertion of another antecedent for "it". So "it (minutes to noon)" would never cross the mind of most readers, in favor of "it (the time 68 minutes ago)". $\endgroup$ – Daniel Wagner Jul 4 '17 at 2:40
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Main answer

First, I'd like you to take a look at the example sentences from the two links you provided (1, 2):

Jack has 3 times as many sweets as John.

Sheila has 5 times as many markers as dave

Now, compare those to the sentence fragment you're trying to interpret:

$x$ is 3 times as $y$

Notice anything different? The key difference here is that, in the examples "as" occurs twice, but in your fragment there is only one "as".

In fact, your fragment (assuming, as you are, that $x$ and $y$ are nouns) is not grammatical in English. It doesn't make sense. So we must look for another interpretation.

The construction in English works as follows:

(nounA) is 3 times as (quality) as (nounB).

This is the form used in both your links. As I've mentioned, there are two instances of the word "as". In other languages we use different words for these two functions, so let's label them for clarity:

(nounA) is 3 times as1 (quality) as2 (nounB).

The first, "as1", introduces a quality that is being compared (how tall, how heavy, how many markers, how many minutes past ten etc.), while the second, "as2", introduces what is being compared - the "baseline" for the comparison.

But, I hear you ask, there is only one "as" in the puzzle - so is it as1 or as2?

Good question!

In English, in this sort of sentence, it is permissible to have just one "as" but only when that is an as1.* In this case, what we are comparing against is unstated and implicit - it must be inferred from the context, and is usually "the thing we are currently talking about". For example:

  • Fred is 3 times as tall. (As me? As a normal person? As he was yesterday? Depends what we're talking about.)
  • Jack has 3 times as many sweets. (As John? As me? As he did yesterday? Depends what we're talking about.)

It is not possible to have just an as2. You simply cannot say:

  • *The cat is 3 times as the dog.
  • *Fred is 3 times as Jack.

It doesn't make any sense; it is ungrammatical. So we can't read the puzzle like that - we must pick a different interpretation.

So:

  • The quality being compared is "number of minutes past 10 a.m." - call that $y$
  • It is three times something (so $y = 3z$)
  • We don't immediately know what it is being compared with. So we don't know $z$; we need to work it out from the context.

Notice that the two clauses in the puzzle are somewhat parallel, in that they share a number of corresponding elements: $$\begin{array}{|c|c|c|} \hline \hbox{(now)} & \hbox{How many minutes?} & \hbox{is it} & \hbox{before 12 noon} \\ \hline \hbox{68 minutes ago} & \hbox{three times as many minutes} & \hbox{it was} & \hbox{past 10 a.m.} \\ \hline \end{array}$$ This is important, because the parallelism gives a clue to the reader to understand elements in the second half in a similar way to how they were understood in the first half.

In particular, the "many minutes" in both parts gives a clue that this is what is being compared. This makes sense because, since it's what the puzzle's question asks, it is what we're currenly talkign about. In short, $z=y$.

(In fact there is a possibility, though it is less likely given the parallelism, that the puzzle intends you to compare the number of minutes since 10 a.m. as of 68 minutes ago with the number of minutes since 10 a.m. now – but @Rob has shown in his answer that this interpretation is impossible from a mathematical point of view.)

* It is possible to have just an as2 in other types of sentence, but not in sentences of the form "X is 3 times as Y".

Other stuff

Here are some other notes on the puzzle that you may find helpful - there are a few other pointers that confirm this interpretation above any other:

The first thing to note is that the question is made up of two main clauses, joined by "if": * How many minutes is it before 12 noon? * 68 minutes ago it was three times as many minutes past 10 a.m.

How many minutes is it before 12 noon?

Concentrating on the first clause: this is simple enough, and you have interpreted it correctly (it is asking for $x$ where the current time is $x$ minutes before 12 noon). But do note the following:

  • Clue: the verb is "is", which indicates that it is asking about the present. There's no specific word to say "now", but none is needed.
  • The subject is "it". This is an impersonal construction; the pronoun "it" doesn't really refer to anything specific: it just refers to the general context. In this case, however, we can replace it with "the time" without changing the meaning.

if

So far a question has been posed (What is $x$?), but we don't have enough information to find it. The "if" signals that the next clause will give further constraints that may allow us to narrow down the answer. It could equally be replaced with "given that".

68 minutes ago

This is adverb phrase indicating the timeframe for the verb. Clue: it indicates that the context for the verb is a specific time in the past - in fact, 68 minutes into the past. Note that this phrase is not the subject of the sentence - though in another context it could be - because it is followed by "it" (the actual subject). Unlike in some languages, English does not allow inserting a subject pronoun that refers to an already-mentioned subject in the same clause (we don't say "*the dog it barked"), so "68 minutes ago" cannot be the subject.

it

As already mentioned, this is the subject. Again, this is an impersonal construction - "it" refers to the context of the verb and (as already established) this means the same as "the time" [at the time of the verb].

was

The verb. Clue: this is past tense, so must be referring to a time in the past. Unlike some languages, for factual "if" clauses such as this, the tense in English is natural - so a past tense implies a reference to the past, whereas in other languages a present tense interpretation is also possible. (Note that things are different for counterfactual "if" clauses.) This is consistent with the fact that, as already mentioned, the timeframe of this verb has been established as "68 minutes ago".

three times as many minutes past 10 a.m.

This is the complement of the verb "was", telling us what the time was in the timeframe of the verb, as a number of minutes past 10 a.m. How many minutes? Let's call that $y$. What is $y$? We don't know exactly, but it is "three times as many minutes". As discussed above, this is an as1, and a number of things point to the implied object of comparison being $x$. So $y=3x$.

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I believe the interpretation the question setter wanted was:

How many minutes is it before 12 noon if 68 minutes ago it was three times as many minutes past 10 a.m. as how many minutes are currently left to 12 noon?

Therefore, if x is the number of minutes to noon then 4x+68=120 so x=13. The only reason I am interpreting it this way is working backwards from the answer. The chances of anybody interpreting the question this way are tiny.

enter image description here
(from xkcd, of course)

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  • $\begingroup$ 4x + 68 = 120 $\rightarrow$ 120 - (x + 68) = 3x This is equivalent to saying: the minutes past 10 am, 68 minutes ago, equal 3 times as many minutes before 12 noon. $\endgroup$ – Samha' Jul 3 '17 at 2:48
  • $\begingroup$ @Samha' I'm not sure if you're disagreeing or agreeing with the answer $\endgroup$ – Rob Jul 3 '17 at 3:50
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    $\begingroup$ This was exactly the way (and the only way) I interpreted the question. $\endgroup$ – Kruga Jul 3 '17 at 7:21
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    $\begingroup$ "The chances of anybody interpreting the question this way are tiny." - This is exactly how I interpreted it. I personally find it really hard to follow the OPs logic in any of their three examples. $\endgroup$ – Chris Jul 3 '17 at 11:02
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    $\begingroup$ @Wen1now The main point here is attribution. On the page you link, they write that the comic comes from xkcd, while you don't. $\endgroup$ – Federico Poloni Jul 3 '17 at 12:33
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How many minutes is it before 12 noon if 68 minutes ago it was three times as many minutes past 10 a.m.?

Lets break down the question.

How many minutes is it

We have to find an unknown amount of minutes. Lets call this x.

before 12 noon

Now we have our first time, which is x minutes before 12.

if 68 minutes ago it was

There is a second time, which is 68 minutes earlier than the first time.

three times as many minutes

Three times as many as what? This could refer to the original x minutes, or the 68 minutes just mentioned. It doesn't make sense to me to use 68, which mean this refers to 3x minutes.

past 10 a.m.?

Our second time is 3x minutes past 10.

To make an equation I will use 10 am as a base time. From 10 am to the first time there are $120 - x$ minutes. From 10 am to the second time there are 3x minutes, then another 68 minutes to the second time for $3x + 68$ minutes.

$120-x=3x+68\Rightarrow x=13$

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  • $\begingroup$ So, how would you accommodate to this interpretation for "times as many" $\endgroup$ – Samha' Jul 3 '17 at 8:25
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    $\begingroup$ @Samha' I don't see any different interpretation of "times as many". That question discusses the ambiguity of "times more", which is not used here. $\endgroup$ – Kruga Jul 3 '17 at 8:39
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    $\begingroup$ The example used in that link is "A has 3 times as many sweets as B, and where B has 5 sweets, then, A has 3 $\times$ 5 = 15 sweets". And that's is the meaning I get to understand up till now, which is the converse of yours. $\endgroup$ – Samha' Jul 3 '17 at 9:01
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    $\begingroup$ @Samha' ok, I finally see what you mean. The problem isn't with "times as many" but because the question doesn't specify if it's "A is 3 times as many as B" or "B is 3 times as many as A". I've been staring at the question for 15 min now, and I still don't think your interpretation is valid, at least without adding a word, like "three times as many as minutes past 10". Even if your interpretation is correct as it is, it would be a very odd way to phrase it. $\endgroup$ – Kruga Jul 3 '17 at 9:43
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    $\begingroup$ @Samha' Huh? How is that the converse? Both usages align with each other. B = minutes before noon, A = minutes after 10, which is 3 × B. $\endgroup$ – David Liu Jul 3 '17 at 19:28
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I think the trouble you are having is not with the interpretation of "times as many" but with what you think the second "it" refers to. The "it" is somewhat abstract, like "it is raining", or "it is 5 o'clock". It does not refer to anything specific, but is a placeholder for the conditions of that particular time and place.

The final part of the question is saying "it was y minutes past 10 o'clock". That 'it' does not refer to a specific number of minutes, but to what the conditions were 68 minutes ago.

So the question should be read as: It is now x minutes before noon, and what is x if 68 minutes ago it was 3x minutes past 10am?

This leads to the answer x=13 as others have noted.

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  • $\begingroup$ A small nitpick, while we don't conciously think of what "it" is, it's actually quite concrete in most cases. It* is often helpful to ask the question to which the statment answers: + What is the weather like? - it (weather) is raining. + What time is it? - it ( the time) is 5 O'clock + What kind of facts do we know about crocodiles? - it (this fact) is well known that crocodiles can exit the river. (*) the process of asking. $\endgroup$ – mr23ceec Jul 3 '17 at 14:33
  • $\begingroup$ @mr23ceec: I disagree: you could perfectly well say "last night it rained", but saying "*last night the weather rained" would be very odd. It's just an impersonal verb. $\endgroup$ – psmears Jul 3 '17 at 21:40
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Here is my sample solution

720 - x - 68 = x * 3 + 600
720 - 600 - 68 = x * 3 + x
52 = 4x
52 / 4 = x
x = 13

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  • $\begingroup$ You've just retyped the equations of the other two answers. Can you extract the flaws in my solutions and justify your interpretation?, respecting my backed up interpretation. $\endgroup$ – Samha' Jul 3 '17 at 7:44
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It's the second option that you mentioned

… if 68 minutes ago it (now) was three times as many minutes past 10 am (then)

x = 3 × [120 - (x + 68)] → x = 39 min

But you're multiplying the wrong thing. The right side of the equasion is X3 larger, so you should write

3 × X = 120 - (X + 68)
4X = 52
X = 13
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There are two interpretations of this question that I can see:

How many minutes is it before 12 noon if 68 minutes ago it was three times as many minutes past 10 a.m.?

  1. The number of minutes between 10am and 68 minutes ago, equals three times the number of minutes from now until noon

  2. The number of minutes between 10am and now, equals three times the number of minutes between 10am and 68 minutes ago.

Now, the second one is interesting, because it opens up the possibility of '10am' being deliberately ambiguous. It could refer to 10am yesterday. Let's investigate that possibility.

For the second case to be true, we have two options:

  1. Both cases refer to the same 10am
  2. 68 minutes ago goes past 10am, and thus each point in time refers to a different 10am.

The first cannot be true. A time closer to 10am (68 minutes ago) cannot also be further away from 10am (3x).

The second cannot be true, either. Three times the number of minutes means we can have a maximum difference of 204 minutes (68 * 3) between the two 10am's. Clearly, that's not the case.

So we've eliminated ambiguity #2. Thus, the question must be:

The number of minutes between 10am and 68 minutes ago, equals three times the number of minutes from now until noon

Which is solved by (let y = the number of minutes past 10am, and x = the number of minutes from now until noon):

y = 120 - (x + 68)
x = y / 3

x = (120 - (x + 68)) / 3
3x = 120 - x - 68
3x = 52 - x
4x = 52
x = 13

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  • $\begingroup$ You have said "... equals three times the number of minutes between 10am ...", but x = y / 3 means "Number of minutes from now until noon, equals one third of the number of minutes between 10am and 68 minutes ago" $\endgroup$ – Samha' Jul 3 '17 at 5:13
  • $\begingroup$ @Samha' Sorry, yes - I had it backwards in the text. I've edited $\endgroup$ – Rob Jul 3 '17 at 5:22
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    $\begingroup$ Regardless of my disagreeing, you didn't edit the part: "The number of minutes from now until noon, equals three times the number of minutes between 10am and 68 minutes ago" $\endgroup$ – Samha' Jul 3 '17 at 5:53
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    $\begingroup$ @Samha' That's untrue - I fail to see any other legitimate interpretation of the question; there's only two possibilities, and we can prove one of them to be unsolvable $\endgroup$ – Rob Jul 3 '17 at 5:59
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    $\begingroup$ @Samha' Well that's just plain incorrect. 'What time is it?' Is a perfect example. Your question is really turning into an English language question instead of a puzzling question. $\endgroup$ – Rob Jul 3 '17 at 22:50
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Well yeah, the text itself doesn't specify exactly that it is right now... However my first idea still was x = (120 -x -68)/3, which would solve to 13.

120 is the total time difference.

x is the time we actually need to know

-x is the amount we are away from 12

-68 are the minutes which makes

/3 one third of the minutes now.

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