Find the values of **A**, **B**, **C**, and **D**
given the following information:

1. **A**, **B**, **C**, and **D** are all positive
integers between 1 and 9.

2. **A**^{3} + B^{3} = C^{3} - D^{3}

3. **C = B + D**

**ANSWER**: There are two solutions:

*A = 6, B = 8, C = 9, D = 1* and

A = 6, B = 1, C = 9, D = 8

**EXPLANATION**: Replacing

**C** with (

**B** +

**D**)
yields:

**A**^{3} + B^{3} = (B + D)^{3} - D^{3}
**A**^{3} + B^{3} = (B + D)(B + D)(B + D) - D^{3}
This ultimately reduces to

**A**^{3} = 3BD(B + D). Since

**C = B + D**,
this can be rewritten as

**A**^{3} = 3BCD. So the
cube of

**A** is divisible by 3. The only possible values of

**A** are 3, 6, or 9
(none of the other single digit integers cubed are divisible by 3). We know from the third piece of
information in the question

**(C = B + D)** that

**B**,

**C**, and

**D** each have unique values as they are all non-zero and

**C** is the highest
of the three. Now, back to

**A**^{3} = 3BCD. If

**A** was 3, the
equation reduces to 9 =

**BCD** and there aren't three

__different__ integers between 1
and 9 that when multiplied together have a product of 9 so we know that

**A** is NOT 3. If

**A** was 9, the equation reduces to 243 =

**BCD** and there aren't three

__different__ integers between 1 and 9 that when multiplied together have a product of 243 so we know
that

**A** is NOT 9. If

**A** was 6, then the equation would reduce to 72 =

**BCD**. There ARE three

__different__ integers between 1 and 9 that satisfy the
equation, namely 1, 8 and, 9. Earlier it was shown that

**C** is the largest of those three
variables so

**C** = 9. Either

**B** = 8 and

**D** = 1 or

**B** = 1 and

**D** = 8.