Problem 1274

1. Let Phillip's number be $a$ and Paula's number be $b$. According to the problem, Phillip's number is greater than Paula's number by 12. Therefore, we can write:
$$a = b + 12$$

2. They each square their numbers and the sum of these new numbers is half of 169. Therefore:
$$a^2 + b^2 = \frac{169}{2}$$

3. Substitute $a = b + 12$ into the equation:
$$(b + 12)^2 + b^2 = \frac{169}{2}$$

4. Expand and simplify the equation:
$$(b^2 + 24b + 144) + b^2 = \frac{169}{2}$$
$$2b^2 + 24b + 144 = \frac{169}{2}$$

5. Multiply through by 2 to clear the fraction:
$$4b^2 + 48b + 288 = 169$$

6. Rearrange the equation to standard quadratic form:
$$4b^2 + 48b + 119 = 0$$

7. Factor the quadratic equation:
$$(2b + 7)(2b + 17) = 0$$

8. Solve for $b$:
$$2b + 7 = 0 \quad \text{or} \quad 2b + 17 = 0$$
$$b = -3.5 \quad \text{or} \quad b = -8.5$$

9. Since $a = b + 12$, calculate $a$ for each $b$:
- If $b = -3.5$:
$$a = -3.5 + 12 = 8.5$$
- If $b = -8.5$:
$$a = -8.5 + 12 = 3.5$$

10. Verify which pair satisfies the condition that Phillip's latest number is greater than Paula's by 5070 when squared again:
- For $b = -3.5$ and $a = 8.5$:
$$a^4 = (8.5)^4 = 5220.0625$$
$$b^4 = (-3.5)^4 = 150.0625$$
$$a^4 - b^4 = 5220.0625 - 150.0625 = 5070$$
- For $b = -8.5$ and $a = 3.5$:
$$a^4 = (3.5)^4 = 150.0625$$
$$b^4 = (-8.5)^4 = 5220.0625$$
$$b^4 - a^4 = 5220.0625 - 150.0625 = 5070$$

Since both pairs satisfy the condition, we can use either pair. The sum of their original numbers is:
$$a + b = 8.5 + (-3.5) = 5$$

The final answer is $\boxed{5}$