http://primepuzzle.com/tunxis/mary's-age.html

Ref: http://www.yofx.org/?p=728

The combined ages of Mary and Ann are 44 years, and Mary is twice as old as Ann was when Mary was half as old as Ann will be when Ann is three times as old as Mary was when Mary was three times as old as Ann. How old is Mary?

Answer: Mary is 27.5

Proof:

This problem can be reduced to 8 equations in 8 unknowns.

There are 4 time periods: the present time (subscript 0) and 3 other past / future times (subscripts 1 thru 3).

m0+a0 = 44                        (The combined ages of Mary and Ann is 44.)
m0 = 2*a1                         (Mary is twice as old as Ann was)
m1 = (1/2)*a2   (or 2*m1 = a2)    (when Mary was 1/2 as old as Ann will be)
a2 = 3*m3                         (when Ann is 3 times as old as Mary was)
m3 = 3*a3                         (when Mary was 3 times as old as Ann.)

a1-a0 = m1-m0                     (The difference between Mary's ages) 
a2-a1 = m2-m1                     (equals the difference between Ann's ages)
a3-a2 = m3-m2                     (for corresponding time intervals.)

The 8 equations above can be written as a matrix equation.

[1  0  0  0  1  0  0  0]  [m0]   [44]
[1  0  0  0  0 -2  0  0]  [m1]   [ 0]
[0  2  0  0  0  0 -1  0]  [m2]   [ 0]
[0  0  0 -3  0  0  1  0]  [m3] = [ 0]
[0  0  0  1  0  0  0 -3]  [a0]   [ 0]
[1 -1  0  0 -1  1  0  0]  [a1]   [ 0]
[0  1 -1  0  0 -1  1  0]  [a2]   [ 0]
[0  0  1 -1  0  0 -1  1]  [a3]   [ 0]

The above matrix was inverted using the online facility at

http://www.bluebit.gr/matrix-calculator/

[0.625 -0.250 -0.250 -0.250  0.375  0.625  1.125  1.125]  [44]   [m0]
[0.563 -1.125 -0.625 -0.625  0.938  0.563  2.813  2.813]  [ 0]   [m1]
[1.375 -2.750 -2.750 -1.750  2.625  1.375  6.875  7.875]  [ 0]   [m2]
[0.375 -0.750 -0.750 -0.750  0.625  0.375  1.875  1.875]  [ 0] = [m3]
[0.375  0.250  0.250  0.250 -0.375 -0.625 -1.125 -1.125]  [ 0]   [a0]
[0.313 -0.625 -0.125 -0.125  0.188  0.313  0.563  0.563]  [ 0]   [a1]
[1.125 -2.250 -2.250 -1.250  1.875  1.125  5.625  5.625]  [ 0]   [a2]
[0.125 -0.250 -0.250 -0.250 -0.125  0.125  0.625  0.625]  [ 0]   [a3]

0.625*44 - 0.250*0 - etc. = m0

m0 = 27.5

Note: for a somewhat related page, go to http://primepuzzle.com/tunxis/solve.html