[judofyr]

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.

[natpalmer1776]

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.

[gnodar]

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?

[quuxplusone]

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.”)

[tempfile]

> 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.

[losvedir]

> 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".

[SV_BubbleTime]

> Since time progresses equally for both of them,

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

[seemaze]

That's pretty much how I approached it;

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

[judofyr]

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.

[stavros]

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.

[JKCalhoun]

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).

[mayoff]

> (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”.

[quickthrowman]

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.