Step 1: Let s = x + y and d = x - y, and write both given facts in terms of s and d.
Since \( x^2 - y^2 = (x+y)(x-y) = sd \), the given difference of squares gives \[ sd = 891 \] Also, \( s^2 + d^2 = (x+y)^2 + (x-y)^2 = 2(x^2+y^2) \), so the given sum of squares gives \[ s^2 + d^2 = 2 \times 3341 = 6682 \]
Step 2: Combine these to find s + d and s - d.
\[ (s+d)^2 = s^2 + 2sd + d^2 = 6682 + 2(891) = 8464 \] \[ s + d = \sqrt{8464} = 92 \] \[ (s-d)^2 = s^2 - 2sd + d^2 = 6682 - 1782 = 4900 \] \[ s - d = \sqrt{4900} = 70 \]
Step 3: Solve for s and d, then for x and y.
Adding the two results: \( 2s = 92 + 70 = 162 \), so \( s = 81 \). Subtracting: \( 2d = 92 - 70 = 22 \), so \( d = 11 \). Since \( s = x + y = 81 \) and \( d = x - y = 11 \): \[ x = \frac{81+11}{2} = 46, \quad y = \frac{81-11}{2} = 35 \] So the two numbers are \[ \boxed{35 \text{ and } 46} \]