Grace, Helen, and Mary

Solution

Grace is 23. 
Her statements are as follows:
I am twenty-two. <False>
I am two years younger than Helen. <True>
I am a year older than Mary. <True>

Helen is 25. 
Her statements are as follows:
I am not the youngest. <True>
Mary and I are three years apart. <True>
Mary is twenty-five. <False>

Mary is 22. 
Her statements are as follows:
I am younger than Grace. <True>
Grace is twenty-three. <True>
Helen is three years older than Grace. <False>


There are different ways that can lead to the right answer. To illustrate their ideas we chose some of your messages and show them below.

Solution by Horst Karaschewski

Mary is 22, Grace is 23 and Helen is 25

This solution can be found by assuming, that Grace's first statement is a lie, hence she's not 22, but her other two statements are true. This gives sequence and distance of the ages. Since Helen is not 3 years older than Grace under our assumption, the first two statements are true, which gives the age of Grace and thus of all the sisters.
 

Solution by Dan Norton

Here is my solution to the Grace, Helen and Mary puzzle.

To start with, there are two direct contradictions: 
A) Grace's age (Grace says 22, Mary says 23) 
B) The relationship between Helen's age and Grace's age (Grace says Helen=Grace+3, Mary says Helen=Grace+2)

Since these contradictions are between the same two people, if Grace is truthful in A then Mary must be truthful in B or if Mary is truthful in A then Grace must be truthful in B.

Hypothesize that Grace is truthful in A and Mary is truthful in B.

Grace is 22 (G)
Helen is Grace+3 or 25 (M)
Mary is Grace-1 or 21 (G)

Now look at the original statements

Grace:
I am twenty-two. True
I am two years younger than Helen. False
I am a year older than Mary. True

Helen: 
I am not the youngest. True
Mary and I are three years apart. False
Mary is twenty-five. False

Mary: 
I am younger than Grace. True
Grace is twenty-three. False
Helen is three years older than Grace. True

Problem: This makes two of Helen's statements false - 
1) Mary(21) and Helen(25) are three years apart. 
2) Mary(21) is 25.

Hypothesize that Mary is truthful in A and Grace is truthful in B.

Grace is 23 (M)
Helen is Grace+2 or 25 (G)
Mary is Grace-1 or 22 (M)

Now look at the original statements

Grace:
I am twenty-two. False
I am two years younger than Helen. True
I am a year older than Mary. True

Helen: 
I am not the youngest. True
Mary and I are three years apart. True
Mary is twenty-five. False

Mary: 
I am younger than Grace. True
Grace is twenty-three. True
Helen is three years older than Grace. False

This solution is correct.

Solution by Michael Graae

Mary is 22. Her statements are true - true - false.
Grace is 23. Her statements are false - true - true.
Helen is 25. Her statements are true - true - false.

The clue is that Grace states she is 22 and Mary states that Grace is 23. If the first is true the second is false and vice versa. If Grace is 22 - then Mary's two other statements are true. This doesn't work out. So Grace must be 23 (and lying about her own age...) - and the puzzle is solved.

Solution by Tina Nolte

Mary is 22, Grace is 23, and Helen is 25.

Let's first check that the answer is correct:

Grace's true statements are that she is 2 years younger than Helen and one year older than Mary.

Helen's true statements are that she is not the youngest and that she and Mary are three years apart.

Mary's true statements are that she is younger than Grace and Grace is 23.

How did I get this answer?
Consider if Grace's statement that she was 22 were true... then according to her statement Helen would be 24. However, Mary's statement that Grace is 23 is false, making her statement that Helen is 3 years older than Grace true. But that makes Helen both 24 and 25. So we know that Grace's first statement is the untrue one. This allows us to determine that Mary's untrue statement is that Helen is three years older than Grace since we know that Grace is two years younger than Helen. Since Grace is 23, we can use the true statements of Grace to immediately conclude that Mary is 22 and Helen 25.

Solution by Trevor Graham

Grace is 23, Helen is 25, Mary is 22.

Let us describe each statement according to the following key...
Grace: 
G1 - I am twenty-two. 
G2 - I am two years younger than Helen. 
G3 - I am a year older than Mary.

Helen: 
H1 - I am not the youngest. 
H2 - Mary and I are three years apart. 
H3 - Mary is twenty-five.

Mary: 
M1 - I am younger than Grace. 
M2 - Grace is twenty-three. 
M3 - Helen is three years older than Grace. 

Removing one statement from each gives 27 different combinations as follows
1. G1, G2 - H1, H2 - M1, M2
2. G1, G2 - H1, H2 - M1, M3
3. G1, G2 - H1, H2 - M2, M3
4. G1, G2 - H1, H3 - M1, M2
5. G1, G2 - H1, H3 - M1, M3
6. G1, G2 - H1, H3 - M2, M3
7. G1, G2 - H2, H3 - M1, M2
8. G1, G2 - H2, H3 - M1, M3
9. G1, G2 - H2, H3 - M2, M3
10. G1, G3 - H1, H2 - M1, M2
11. G1, G3 - H1, H2 - M1, M3
12. G1, G3 - H1, H2 - M2, M3
13. G1, G3 - H1, H3 - M1, M2
14. G1, G3 - H1, H3 - M1, M3
15. G1, G3 - H1, H3 - M2, M3
16. G1, G3 - H2, H3 - M1, M2
17. G1, G3 - H2, H3 - M1, M3
18. G1, G3 - H2, H3 - M2, M3
19. G2, G3 - H1, H2 - M1, M2
20. G2, G3 - H1, H2 - M1, M3
21. G2, G3 - H1, H2 - M2, M3
22. G2, G3 - H1, H3 - M1, M2
23. G2, G3 - H1, H3 - M1, M3
24. G2, G3 - H1, H3 - M2, M3
25. G2, G3 - H2, H3 - M1, M2
26. G2, G3 - H2, H3 - M1, M3
27. G2, G3 - H2, H3 - M2, M3

The following linked statements are incompatible,

1. G1 ("I am twenty two") and 
M2 ("Grace is twenty three") 
thus options 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18 are invalid.

2. G2 ("I am two years younger than Helen") and 
M3 ("Helen is three years older than Grace") 
thus options 2, 3, 5, 6, 8, 9, 20, 21, 23, 24, 26, 27 are invalid.

3. G1 ("I am twenty-two"), 
G3 ("I am a year older than Mary") and 
H3 ("Mary is twenty-five") 
thus options 13, 14, 15, 16, 17, 18 are invalid

4. H3 ("Mary is twenty-five"), 
M1 ("I am younger than Grace") and 
M2 ("Grace is twenty-three") 
thus options 4, 7, 13, 16, 22, 25 are invalid

5. G3 ("I am a year older than Mary"), 
H2 ("Mary and I are three years apart") and 
M3 ("Helen is three years older than Grace") 
thus options 11, 12, 17, 18, 20, 21, 26, 27 are invalid

The only remaining valid option is number 19, the following statements are therefore true. 

Grace 
"I am 2 years younger than Helen" - 
"I am a year older than Mary"

Helen - 
"I am not the youngest" - 
"Mary and I are three years apart"

Mary - 
"I am younger than Grace" - 
"Grace is 23"

The rest is simple mathematics starting from Mary's declaration about Grace's age and Grace's declarations about the age gaps between her and her friends.