1. **State the problem:** You have 50 lumber to allocate between hp and defense (def). Each lumber gives 160k hp or 640 def. Each point of defense increases hp by 2%. 2. **Define variables:** Let $x$ be the lumber allocated to hp, and $y$ be the lumber allocated to def. So, $x + y = 50$. 3. **Calculate base hp:** If you spend $x$ lumber on hp, base hp is $$\text{HP}_{base} = 160000 \times x.$$ 4. **Calculate defense points:** If you spend $y$ lumber on defense, defense is $$\text{DEF} = 640 \times y.$$ 5. **Calculate total hp increase from defense:** Each def increases hp by 2%, so total multiplier is $$1 + 0.02 \times \text{DEF} = 1 + 0.02 \times 640y = 1 + 12.8y.$$ 6. **Calculate total hp:** $$\text{HP}_{total} = \text{HP}_{base} \times (1 + 12.8y) = 160000x(1 + 12.8y).$$ 7. **Rewrite using $y = 50 - x$:** $$\text{HP}_{total} = 160000 x (1 + 12.8 (50 - x)) = 160000 x (1 + 640 - 12.8 x) = 160000 x (641 - 12.8 x).$$ 8. **Expand:** $$\text{HP}_{total} = 160000 (641x - 12.8 x^2) = 160000 \times 641 x - 160000 \times 12.8 x^2 = 102560000 x - 2048000 x^2.$$ 9. **Maximize $\text{HP}_{total}$ as a quadratic function:** It’s a parabola opening downwards with vertex at $$x = -\frac{b}{2a} = -\frac{102560000}{2 \times (-2048000)} = \frac{102560000}{4096000} = 25.$$ 10. **Allocate lumber:** To maximize total hp, allocate 25 lumber to hp and 25 lumber to def. 11. **Final total hp:** $$\text{HP}_{total} = 102560000 \times 25 - 2048000 \times 25^2 = 2564000000 - 1280000000 = 1284000000.$$ So total hp is 1,284,000,000. **Answer:** Allocate 25 lumber to hp and 25 lumber to defense to maximize hp.