How old is Ann?

> 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.

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

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.

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 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.