Diamond Problem Calculator

What is a diamond problem?

A diamond problem arranges four numbers in a diamond shape: the top cell holds the product of two factors, the bottom cell holds their sum, and the left and right cells hold the factors themselves. Given any two of those four numbers, the other two are fixed — solving the puzzle is really just solving a small system of equations.

How the calculator solves it

If you know both factors, it multiplies and adds them directly. If you know one factor plus the product or sum, it uses simple algebra to isolate the missing factor. If you only know the product and sum, it treats the two factors as roots of x² − (sum)x + (product) = 0 and applies the quadratic formula — or you can use our Quadratic Equation Calculator — which is exactly the skill diamond problems are meant to build toward factoring trinomials.

Reading negative and fractional results

A negative product means the two factors have opposite signs; a negative sum with a positive product means both factors are negative. When a product-and-sum pair has no real factor pair — the discriminant sum² − 4×product is negative — the calculator will say so rather than force an answer.