Key idea. Equal sums and equal sums of squares force equal products.

1. Write the common sum as \(s=a+b=c+d\). Squaring gives

   \[s^2=a^2+b^2+2ab=c^2+d^2+2cd.\]

2. The sums of squares are equal by the question. Subtract them from the equality above to obtain \(2ab=2cd\), so \(ab=cd\). Call this common product \(p\).

3. From the cube identity,

   \[a^3+b^3=(a+b)^3-3ab(a+b)=s^3-3ps.\]

4. Exactly the same calculation gives \(c^3+d^3=s^3-3ps\). Hence the two sums of cubes are equal.

Remember. Two pairs with the same sum and product have the same sum of cubes.