> (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.”)
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.
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.
"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.
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.)
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.
> 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.
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.
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.
> 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.
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".
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.
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.
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.
> 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.
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:
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.
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.
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.
"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.
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...
Couple of minutes to understand the english, and an editor to write the formulas :-)
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")
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).
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.
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.
Wait so how old is she?
well into the post-geriatric era by now
18
I am surprised no one spotted the obvious solution:
perl -e '$_=24;print+(y///c,$m=$_)&&$_*3/4 .$/'
> 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.
Here's the simplest form I could distill it to. The other comments I find quite verbose or confusing, no offense intended.
That's not how I read it. I read it as:
Moreover, we have x + delta = 24, and 12 + delta = x (they get older at same rate), so delta = 6 and x = 18.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:
Which many of us can solve in our heads without writing down, or even without consciously converting "Ann's current age" to "X".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.
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.
This has flashbacks to GMAT problems.
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.
Opus says Ann is 18
Sounds about right.
How I understand it:
Mary is 24, Ann is a few (n) years younger
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: 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
and the circle remains unbroken
But that was over a hundred years ago. She must be dead by now.