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

1. **A**, **B**, **C**, and **D** are all unique
positive integers between 1 and 9 (no two variables have the same value).

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). 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.

**A** must be 6 therefore and the equation reduces to 72 = BCD. The only three

__different__ integers between 1 and 9 that when multiplied together make 72 are 1, 8, and 9 OR 2, 4
and 9. However, only 1, 8, and 9 can satisfy the equation

**C** =

**B** +

**D** (i.e. where the highest number is the sum of the other two).

**C** is the
largest of the three, so

**C** = 9. Either

**B** = 8 and

**D** = 1
or

**B** = 1 and

**D** = 8.