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.

> 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