Two boys are playing on a ground, and both are less than 10 years old. The age of the younger boy is equal to the cube root of the product of the ages of the two boys. If we place the digit representing the age of the younger boy to the left of the digit representing the age of the elder boy, we obtain the age of the younger boy's father. Similarly, if we place the digit representing the age of the elder boy to the left of the digit representing the age of the younger boy and then divide that two-digit number by 2, we obtain the age of the younger boy's mother. The mother is younger than the father by 3 years. What are the ages of the elder and younger boys?

Introduction / Context:
This is a logical ages puzzle involving two young boys and their parents. The puzzle links the ages of the boys with their parents' ages using a combination of digit manipulation and a cube root condition. Questions like this test both numerical reasoning and the ability to interpret word problems carefully.

Given Data / Assumptions:
Both boys are less than 10 years old. The age of the younger boy equals the cube root of the product of the ages of the two boys. When the digit of the younger boy's age is placed in front of the digit of the elder boy's age, the resulting two-digit number gives the age of the younger boy's father. When the digit of the elder boy's age is placed in front of the digit of the younger boy's age and the resulting number is divided by 2, we obtain the age of the younger boy's mother. The mother is 3 years younger than the father. We must determine the ages of the elder boy (E) and younger boy (Y).

Concept / Approach:
Since both boys are less than 10 years old, their ages are single-digit positive integers. The cube root condition says that Y is the cube root of E * Y. This implies Y^3 = E * Y, and because Y is not zero, we can divide both sides by Y to get Y^2 = E. Thus, E must be the square of Y. We then test small integer values of Y to find suitable E values that remain single-digit and check whether the parent age conditions are satisfied.

Step-by-Step Solution:
Step 1: From Y^3 = E * Y, divide by Y (Y is positive and nonzero) to get Y^2 = E.Step 2: Try small values of Y. If Y = 1, then E = 1^2 = 1, but then there is no meaningful elder and younger distinction, so discard.Step 3: If Y = 2, then E = 2^2 = 4, which is valid and less than 10.Step 4: If Y = 3, then E = 9, which is not among the options and would lead to different parent ages. Also, options explicitly list specific pairs.Step 5: Now test Y = 2 and E = 4 with the digit conditions. Placing the digit of the younger boy (2) before the elder boy (4) gives 24. This is the age of the father.Step 6: Placing the digit of the elder boy (4) before the younger boy (2) gives 42. Dividing by 2 gives 21, which is the age of the mother.Step 7: Check the difference between the father and mother: 24 - 21 = 3 years, which matches the condition that the mother is 3 years younger than the father.Step 8: Therefore, E = 4 years and Y = 2 years satisfy all the puzzle conditions.

Verification / Alternative check:
Recheck the cube root condition: E * Y = 4 * 2 = 8. The cube root of 8 is 2, which equals Y, so the first condition holds. The father's age as 24 and mother's age as 21 also satisfy the difference of 3 years. Thus, all relationships are consistent and confirm the solution.

Why Other Options Are Wrong:
The pairs (15, 3), (14, 12), and (40, 22) violate the basic constraints of the puzzle because the boys must both be less than 10 years old and have single-digit ages. Moreover, they do not satisfy the cube root relationship Y^3 = E * Y nor the specific digit-based conditions for the parents' ages. Only E = 4 and Y = 2 meet every requirement.

Common Pitfalls:
Some learners may ignore the restriction that both boys are less than 10 years old and start considering two-digit ages. Others might misinterpret the digit placement or forget that the cube root condition forces a very specific relationship between E and Y. Staying systematic by translating each statement into a clear algebraic or digit condition greatly simplifies the puzzle.

Final Answer:
The elder and younger boys are E = 4 years and Y = 2 years old respectively.