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How old is Ann?

quuxplusone.github.io

71 pointsibobev71 comments

Ann is 18.

I think what makes it confusing is the "variable overloading" referring to both the current ages of Mary and Ann (M & A), and their ages at some point in the past "when Mary was as old as Ann is now" (M' & A').

So, what we're given is:

M = 24

M' = A

M = 2A' => A' = M/2 = 12

Since the age gap between Mary and Ann is constant, we know:

M - A = M'- A'

So, substituting in the known values:

24 - A = A - 12

2A = 36

A = 18

We can double check the result. Since Mary (24) is 6 years older than Ann (18), then when Mary was 18 Ann would have been 12, so Mary is now twice that age as given.

I think it’s easier to see if you drop the time indices and just introduce a variable for the elapsed time. So:

Mary := 24 = 2(A - Δ)

A = Mary - Δ

And then it’s just substitution. But of course what appeals to one’s intuition is very personal

To me, explicitly representing the two pairs of ages (now: M, A; previously: M', A') is simpler because it let's you directly represent the problem statement without any mental gymnastics whatsoever.

  M = 2A' "Mary is [now] twice as old as Ann was ..."

  M'= A   "when Mary was as old as Ann is now"

Thanks for sharing this, it actually makes it easy. I somehow convinced myself it was a trick having to do with ages being rounded off to whole numbers, so delta-age can be off by one... anyway I want my day back.

Rephrasing it makes it easier to grasp:

Mary is 24 years old. When Mary was Ann’s current age, Ann was 12 years old (half Mary’s current age).

This makes it a bit easier to realize that Ann’s age has to be right in the middle of 12 and 24.

Ah, that makes sense. I originally read it differently.

> Mary is 24 years old. She is twice as old as Ann was when Mary was as old as Ann is now. How old is Ann?

Which I read as:

> Mary is 24 years old. When Mary was Ann's current age, Anne was half the age she currently is.

Which would mean Anne is 16 (because when Mary was 24-8=16, Anne's current age, then Anne was 16/2=8, half Anne's current age).

But re-reading it, then for that to be true the original would have needed to be phrased:

> Mary is 24 years old. She was twice as old as Ann was when Mary was as old as Ann is now. How old is Ann?

Indeed,

> (A writer to the Montgomery (Alabama) Advertiser of 1903-10-24 points out that you can get the apparently-most-common wrong answer if you read the puzzle incorrectly as “She was twice as old as Ann was when Mary [sic] was as old as Ann now is.”)

> This makes it a bit easier to realize that Ann’s age has to be right in the middle of 12 and 24.

How did you actually make that deduction? I can only see it by writing down an equation. I can't see anything in the problem that directly implies it.

> How did you actually make that deduction?

Think of it on a number line. Mary is 24 now, Ann was 12 then. There's a point somewhere between the two, representing Mary's age at that time.

     ---*-------*---------------*--------
        12      ?               24

The problem states that time has passed such that Mary has aged to 24 and Ann has aged to that point. You can think of time passing as a line growing out of each of them:

     ---*===----*===------------*--------
        12      ?               24

Since time progresses equally for both of them, and Ann is now that mystery age, and Mary is now 24, you can see that the distance from "12" to "?" has to be the same distance as "?" to "24".

> Since time progresses equally for both of them,

Wild assumption. I saw absolutely nothing in the text to indicate their relative velocities.

That's pretty much how I approached it;

                   A1   M1
              A0   M0
    +----+----+----+----+----+
    0    6    12   18   24

I imagined that I'm 24 years old and I have a 20 years old brother. When I were his age, he would have been 16 years old.

I also realized that this can be expressed in terms of a two pairs of sibling:

> Mary and Ann are the same age difference as Jane and Claire. Mary is 24 years old. Mary is twice as old as Jane. Claire is as old as Ann. How old is Ann?

This highlights also why it's so confusing:

> Mary is 24 years old. She [Mary, today] is twice as old as Ann was [Ann, past] when Mary was [Mary, past] as old as Ann is [Ann, today] now. How old is Ann?

In one sentence we're comparing past and present ages.

Mary is 24 years old. When Ann was 12, Mary was Ann’s current age.

Let's say Ann's current age is 13. Then, when Ann was 12, Mary must have been 13. Now that Ann is 13, Mary must be 14, but she's 24.

This means that the only way for Mary's-age-when-Ann-was-12 to have been Ann's-age-now is for the same amount of years to have passed between Ann being 12 to being Ann's-age-now than from Ann's-age-now/Mary's-age-then to Mary's 24, which is 18.

Personally, I would rephrase it to:

Ann is 6 years younger than Mary, who is 24 years old.

Or even better, Ann is 18 and Mary is 24.

But then again, I always disliked word-based logic problems that didn't correspond to genuine complexity. If something CAN be stated in simple terms, I believe it should be. If you want to create a word-based logic puzzle then present a genuine problem that can be engaged with rather than a poorly formulated explanation of the mundane.

That being said, I have no idea how I would present the same fundamental logical puzzle using a material problem that doesn't have a simpler distillation that solves the problem for you. Maybe I'm just not cut out for the realm of intellectual puzzles.

Without rephrasing it… I determined that Ann is 16.

I'm not sure why it needs rephrasing—it seems pretty straightforward? What am I missing?

16 works (as another pointed out). Ann is 16 and Mary would have been that age 8 years ago—when of course Ann would then have been 8 (Mary, at 16, was twice her age).

> (Mary, at 16, was twice her age)

The problem states “She _is_ twice as old as Ann was”, not “She _was_ twice as old as Ann was”.

I figured it out by knowing that Ann had to be 12 in the past since Mary is 24 now which is twice Ann’s age. 6 years have passed since Ann was 12 and Mary was 18 making them 18 and 24, respectively.

I don't know why but I really struggled to internalize what the question was asking. Here's my take on a rewrite that's clearer:

Mary is 24, which is twice as old as Ann was some time ago. At that time, Mary was the age that Ann is now. What is that age?

This may help people:

> Mary is 24 years old. She is twice as old as Ann was (12) when Mary was as old (18) as Ann is now (18). How old is Ann?

So 6 years ago, Mary was 18, and Ann was 12.

Today, Mary is twice the age that Ann was (12 * 2) at the time that she (Mary) was 18, making her 24.

What makes it confusing is that the sentence is comparing Mary’s present age with Ann’s past age (which I did not catch until mulling it over a bit).

Yeah, I had to spell it out to get it to click for me

> mary is 24

> 8 years apart: ann is 16 now, when ann was 12 mary was 20

> 6 years apart: ann is 18 now, when ann was 12 mary was 18

> 4 years apart: ann is 20 now, when ann was 12 mary was 16

Mary IS 24. So Ann WAS 12. Say that was X years ago. Then Mary WAS 24-X and Ann IS 12+X. So 24-X=12+X, X=6, Ann is 18.

I can't see a way to do it without algebra.

there's 12 years between 12 and 24. it is divided equally between OldAnn-OldMary and OldMary-NowMary. 12/2=6, 12+6=18

It's a kind of cheating because it means not understanding the essence of the problem but with such small numbers enumeration is a viable strategy.

> Mary is 24 years old. She is twice as old as Ann was

So Ann was 12

> when Mary was as old as Ann is now

So Mary is older than Ann.

The age of Ann is somewhere between 12 and 24. Without much thinking I'd say that probably 12 and 24 are not included and probably it is an even number.

We can test each of them.

If Ann is 14 now, when Mary was 14 (10 years ago) Ann was 4 year old and Mary would be 8 now, this is not the solution.

If Ann is 16 now, when Mary was 16 (8 years ago) Ann was 8, Mary would be 16 now, nope.

If Ann is 18 now, when Mary was 18 (6 years ago) Ann was 12, Mary would be 24 now and she is. This is the solution.

Ann is 18.

> It's a kind of cheating because it means not understanding the essence of the problem

In my opinion, this method presumes a clear understanding of the problem itself.

If you didn't have a clear understanding of the problem, you would not be able to test an answer to see if it is correct.

Now, this clear understanding of the problem could, of course, be coupled with a lack of understanding of other methods available (such as algebra) to solve the problem.

OTOH, maybe it's just coupled with a clear understanding that with the working memory that you have available to you right at the moment and without writing anything down, you don't even need those other methods.

It's not cheating, but it's not enlightening either.

> I can't see a way to do it without algebra.

This is related to "When all you have is a hammer, everything looks like a nail."

It's slightly different, because you've supercharged your hammer. Everything you ever understood about how to solve word problems has been subsumed into what your brain labels as "algebra."

But look at it this way:

1) Did you use algebra to convert the problem to algebraic form? Probably not; algebra says nothing about word problems.

2) Once it was in algebraic form, did you need to repeatedly apply algebraic rules in order to reduce the problem, or could you glance at it and figure it out?

You may also be hampered by your choice of variable, because you chose "X" to be an intermediate variable.

If, instead, you choose "X" to be what you are searching for, Ann's current age, then the problem setup is:

  24 - x = x - 12

Which many of us can solve in our heads without writing down, or even without consciously converting "Ann's current age" to "X".

Sorry to say, the first half of this comment is almost gibberish. It became algebra when I introduced an unknown variable and wrote down an equation. I can't see a way to solve the problem without doing that.

> If X is Ann's current age, then the problem setup is: 24 - x = x - 12

How did you get this equation from the problem statement? The equation is of course correct, but I don't see how you would derive it, other than writing down a more obvious equation and rearranging it.

> Sorry to say, the first half of this comment is almost gibberish.

In what way?

> It became algebra when I introduced an unknown variable and wrote down an equation.

But a lot of people, including me, can solve it without writing down any equation.

> I can't see a way to solve the problem without doing that.

Ah. So it's gibberish because of your limitations? That's... not how this usually works. (Although, to be fair, it's often the case that people of limited intellectual means lash out with unkind comments such as "gibberish" so maybe this is how it works.)

> How did you get this equation from the problem statement?

"Mary is 24 years old. She is twice as old as Ann was when Mary was as old as Ann is now. How old is Ann?"

We know that there is a delta between Mary's current age (24) and what I called X in my previous comment (Ann's current age AKA Mary's prior age). That would be 24 - X.

We also know that at some previous time, Ann was 12 (half of 24) when Mary was (Ann's current age AKA Mary's prior age).

Some of us just take the mental shortcut that the in between age has to be the average of 24 and 12 (because the delta doesn't change[1], so the delta must have been the same before as it is now), but if forced to write it down into algebra, we can say that this second statement is X - 12.

And then of course, again, because the delta doesn't change, and the delta is equal to both x-12 and 24-x, those expressions must be equal to each other.

[1] Except of course, the delta could wobble a bit for birthdays being on different dates within the year. In simple puzzles like this, of course, that +/- 1 year possibility is usually ignored.

I'll answer in reverse order.

I think what you suggest works! Putting it in the wording of my original comment: Mary IS 24, Ann WAS 12, and the same amount of time gets you from 12 to Ann's current age, and from Ann's current age to 24. I think it is obvious from there that you're half-way between, so Ann is 18. Thank you!

The part of your earlier comment I consider gibberish has nothing to do with the problem.

> It's slightly different, because you've supercharged your hammer.

This does not mean anything.

> Everything you ever understood about how to solve word problems has been subsumed into what your brain labels as "algebra."

This is bizarre speculation. Why are you analyzing my brain? I just asked for an explanation!

> Did you use algebra to convert the problem to algebraic form?

As you say, this is impossible to do, because it doesn't make sense. So why bring it up? It sounds like you are trying to convince me the problem is not algebra - or that I am in some way only seeing it as algebra because I am not thinking about the problem clearly. But I never said it was! I just said I can't see how to do it without algebra. Since you mixed those things up, I clarified by saying exactly what sense I was talking about algebra in the first place. Clearly that didn't work!

Again, thank you for explaining your reasoning, I appreciate it. Just leave out the analysis of my mind!

You're welcome.

> Just leave out the analysis of my mind!

I should have added weasel words like "maybe" and "probably" because I don't know you. Yet, everything I wrote, I have observed numerous times in other people.

And (and of course, maybe I'm wrong here) if you hadn't learned algebra, this solution would have been more obvious to you. Obviously we can't run that experiment directly. But think back to your childhood. Did you ever intuitively know the answer to a problem that others struggled with? Does that happen as often lately?

Formal methods like algebra are powerful and allow us to document, step by step, transformations that would be impossible to hold in our heads informally. We can solve problems that were unapproachable before. But we can get so used to using them that we forget intuitive tricks that we used to use.

Maybe this didn't happen to you. You're right. I don't know you. I'm only extrapolating. It really is the sort of simple problem that many people (possibly most of them younger) can solve immediately without assigning variable names or even writing anything down.

If this describes your capabilities when you were younger, then something changed. What is it?

Let me give you an example of my own. When I was a child I would play around with electronics. I intuitively knew that if I put two resistors in parallel, the amount of resistance would go down, and by how much, and could easily extend that to 3 or 4 resistors, rummaging through my collection to find resistors that would parallel to give me the value I wanted.

Once I learned the parallel resistor formula, I got slower at everything above two resistors.

Which brings us to:

> As you say, [using algebra to convert the problem to algebraic form] is impossible to do, because it doesn't make sense. So why bring it up?

I brought it up because, although we agree that it is not strictly part of algebra, it is something that you had to learn in order to use algebra effectively.

To me (and of course, again, I'm still speculating here) the fact that you're smart enough to use intuitive methods to set the problem up in algebraic form means that you were probably smart enough to use intuitive methods to simply solve the problem directly, if you weren't so used to directing your brain activity towards doing things using algebra.

Uhhh... what? The OP is right; even if you do it intuitively it's still algebra. If you need a proverb to justify it: Just because your hammer is made of stone doesn't make it less of a hammer.

Thanks for the downvote.

Look, when someone says "you need algebra to solve this" is it reasonable to assume that they are talking about informal methods that people have used forever, or is it more reasonable to assume they are talking about formal algebraic methods?

Because many people sure as shit don't need any algebraic symbols or operators to solve this in their heads.

"Any eighth-grade boy or girl who is 'all there' should solve it in about thirty seconds" is hyperbole (it took me a couple of minutes), but in general the public reaction to this seems like a demonstration of the Flynn effect.

> it took me a couple of minutes

Are you in the 8th grade? Those of us a couple of decades past the 8th grade are maybe slower at algebra than someone who is actively drilling it...

I must be getting old as well because I had to write it down in a piece of paper to do it, but I did get it pretty quick.

This is more a test of reading comprehension than algebra. Figuring out what the questions is even trying to ask is like 90% of the question.

When I was that age, we practiced a lot of word problems. The folks who wrote the tests really favoured this kind of slightly underhanded logic

They aren't allowed to make the mathematics too difficult so they make up for it by making the questions hard to understand.

Couple of minutes to understand the english, and an editor to write the formulas :-)

Not the Flynn effect; just people having fun, and selection bias. I (the blogger) deliberately left out many boring responses, except for linking to the Evening World's letters page of 1903-11-11.[1] I filtered out scores of replies from schoolchildren walking through the algebra to get the correct answer, and dozens of (usually short) replies from people of various ages giving wrong answers with and without reasoning, some almost certainly joking, some almost certainly not. But that's what you should expect from a Letters to the Editor section on any given math meme! Our discourse hasn't changed that much over 120 years. See my other post,[2] linked at the bottom of the current one, on "What is 8÷2(2+2)?" — that's not as fair a question, but the discourse is pretty much the same in unfiltered Letters-to-the-Editor sections: "I think it's X. Well, I think it's Y."

The whole point of this kind of meme is for the populace to perform disagreement about the answer! (See also: the Monty Hall problem; sports; comments sections.)

[1] - https://www.loc.gov/resource/sn83030193/1903-11-11/ed-1/?sp=...

[2] - https://quuxplusone.github.io/blog/2019/08/01/what-is-8-divi...

It's amazing to me how much effort people invest in elaborating wrong answers.

The key insight here is to remember that time moves the same for both Mary and Ann. Let's call the number of years between past and present X. Then

24 - X = 12 + X => 24 - 12 = 2X => X = 6.

Ah yes. The famous problem that is ambiguous because it ignores space travel!

Let M_n := Mary's age now = 24, A_n := Ann's age now, and let M_p, A_p be the ages of Mary and Ann at a certain point in the past. We have that M_n = 24 = 2(A_p) from "Mary is 24 years old. She is twice as old as Ann was", immediately implying that A_p = 12. At the point in time p when M_p, A_p were the ages of Mary and Ann, respectively, we have that M_p = A_n from "when Mary was as old as Ann is now". Also, let x := | M_n - M_p | = | A_n - A_p | be the amount of time that has passed between the point p in time and now. Since M_n > M_p and A_n > A_p by construction, we can drop absolute values, yielding M_n = M_p + x, A_n = A_p + x. Substituting, we have M_p = 24 - x, and A_n = 12 + x, yielding x = 24 - M_p = A_n - 12, yielding A_n = 36 - M_p. But, since M_p = A_n, we can write 2(A_n) = 36 yielding A_n = 18.

This problem, however, completely ignores the fact that Ann boarded an interstellar spaceship 6 years ago at a reasonable fraction of the speed of light c. To account for this missing detail, we need to use Lorentz factors. Since Ann was the one traveling in space, we have to adjust her current age by calculating A_n = A_p + x * sqrt(1 - (v/c)^2)), where v is Ann's velocity Given that the problem is missing the critical detail of how fast Ann is hurtling towards the outer boundary of the universe, we really can't calculate their age at all. Nonetheless, given the completely reasonable and plausible assumption that Ann has been traveling at 95% of the speed of light because spaceships totally can do that, Ann is obviously about 13.87 years old. So 18 years old is clearly the wrong answer.

Note that the fact that Ann is not aging as much due to traveling at an enormous velocity does NOT change Mary's age. Mary remains exactly twice as old as Ann was at the given point in the past, so she's 24 and 6 years have passed from her perspective. Only Ann's age changes. Obviously.

Here's the simplest form I could distill it to. The other comments I find quite verbose or confusing, no offense intended.

    x: Ann's current age
    y: The age difference between Ann and Mary

    24 = x + y      (ie. their current ages)
    x = (24/2) + y  (ie. their age in the past)

    Solve for y in one equation and plug it into the other. You'll get x = 18.

That's not how I read it. I read it as:

  Now : Mary is 24 = 2y,  Ann is x
  Past: Mary is x,        Ann is 12 = y

Moreover, we have x + delta = 24, and 12 + delta = x (they get older at same rate), so delta = 6 and x = 18.

> When Mary had a little lamb, she was twice as old as Ann was when Ann was as old as Mary now claims to be. How old is the sheep? —James Barton Adams.

I love this response so much.

The closing problem is more fun.

> A man is twice the age his wife was when he was the age she is now. When she reaches his present age, their combined years will be 100. Find the age of each.

This seemed initially like something amenable to guessing and checking a few ratios. <narrator voice>It was not.</narrator voice>

My answer: The man is 44 4/9 years old, the wife 33 1/3 years.

I am surprised no one spotted the obvious solution:

perl -e '$_=24;print+(y///c,$m=$_)&&$_*3/4 .$/'

Like all of these things that get posted on facebook regularly, they almost intentionally use vague or improper language to be confusing so realistically there is no valid answer. You have to rewrite the problem in clear, unambiguous language and then they are usually easily solved.

Wait so how old is she?

well into the post-geriatric era by now

Barely legal

My approach at parsing this:

There's two point in time with two ages each:

M, M0, and A, A0

Then the sentence can be expressed as:

* M = M0 + X and A = A0 + X and (X is the time difference)

* M = 24 and ("Mary is 24")

* M = 2 * A0 and ("Mary is twice as old as Ann was")

* M0 = A ("Mary was as old as Ann is now")

Yes, this was exactly what I had, pretty much. A system of equations, 4 equations, 4 variables.

MO (mary old), MN (mary now), AO (ann old), AN (ann now)

MN = 24

MO = AN

2AO = MN

MN - AN = MO - AO

How interesting!

This discussion itself offers insight on the actual need for technical professionals to be able to logically reason through non-obvious [information] structures.

I've anecdotally suspected that the ability to synthesize information into a navigable data structure and subsequently analyze it, both from internal and external POV, is not a hard requirement for the majority SWE positions.

This has flashbacks to GMAT problems.

83. This is an AI verification referring to my aunt. I didn't read the article and skimmed the comments.

what the true answer? oh, there is no? My intuition says that Ann now is 24

> Dozens of people declare solemnly that Mary and Ann are twins, though the slightest attempt to "prove" this shows it does not fulfil the conditions of the problem. [...] "About Mary and Ann, I think they are eight years old," says a woman, disregarding the first words of the question, which state that Mary is twenty-four.

[1] - https://www.loc.gov/resource/sn88085488/1903-10-31/ed-1/?sp=...

18.

Ann is 7 years old.

Opus says Ann is 18

Sounds about right.

How I understand it:

Mary is 24, Ann is a few (n) years younger

  M = 24
  A = M-n

Mary is twice as old as Ann was when Mary was as old as Ann is now. So we have to deduct the age difference twice from Mary's current age to find out how old Ann was when Mary was as old as Ann is now, which is half Mary's current age:

  M-2n = M/2 = 12

  M-2n = 12 --> n=6

So the age difference is 6, and Ann is 18.

mine was not so parsimonious, my teachers would probably say i am not very clever but has points for effort.

ann_old=12

mary_curr=24

mary_old=ann_curr

mary_curr-ann_curr=X

mary_old-ann_old=X

24-ann_curr=X

mary_old-12=X

ann_curr-12=X

24-ann_curr=ann_curr-12

2ann_curr -12 = 24

2ann_curr = 36

ann_curr = 18

For me they key step to solving it was realizing that

mary_curr - mary_old = ann_curr - ann_old

since they must have aged the same amount of years.

time dilation joke incoming

But that was over a hundred years ago. She must be dead by now.

and the circle remains unbroken

Hallucinations like these make it pretty clear that these agents do not really understand or think, IMHO.

"these agents" meaning the people who tried to solve the problem and failed?

It is well known that humans often fail at thinking. Using that as proof that machines can think only proofs that you’re human.

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