Points Combination 89Eed6

1. **State the problem:** We want to find a combination of points from Shards, Heroes, SSR, and SR that sum exactly to $12,062,000$ points. 2. **Define variables and constraints:** Let: - $x$ = number of Shards, each worth $25,000$ points, and $x$ must be a multiple of $5$. - $y$ = number of Heroes, each worth $4,500$ points, minimum $30$. - $z$ = number of SSR, each worth $8,750$ points, multiples of $10$ if used. - $w$ = number of SR, each worth $2,500$ points, multiples of $10$ if used. 3. **Write the equation:** $$25,000x + 4,500y + 8,750z + 2,500w = 12,062,000$$ 4. **Apply constraints:** - $x$ multiple of $5$ - $y \geq 30$ - $z$ multiple of $10$ if $z > 0$ - $w$ multiple of $10$ if $w > 0$ 5. **Approach:** Since this is a Diophantine equation with constraints, we try to find integer solutions satisfying all conditions. 6. **Example solution attempt:** Try $x=480$ (multiple of 5), then: $$25,000 \times 480 = 12,000,000$$ Remaining points: $$12,062,000 - 12,000,000 = 62,000$$ Try $y=30$ (minimum): $$4,500 \times 30 = 135,000$$ This exceeds remaining points, so reduce $x$. Try $x=479$: $$25,000 \times 479 = 11,975,000$$ Remaining: $$12,062,000 - 11,975,000 = 87,000$$ Try $y=30$: $$4,500 \times 30 = 135,000$$ Still too high. Try $y=19$ (less than minimum, invalid), so increase $x$. Try $x=481$: $$25,000 \times 481 = 12,025,000$$ Remaining: $$12,062,000 - 12,025,000 = 37,000$$ Try $y=8$ (less than minimum), invalid. Try $x=480$, $y=0$ (less than minimum), invalid. Try $x=479$, $y=4$ (less than minimum), invalid. Try $x=478$: $$25,000 \times 478 = 11,950,000$$ Remaining: $$12,062,000 - 11,950,000 = 112,000$$ Try $y=30$: $$4,500 \times 30 = 135,000$$ Too high. Try $y=24$ (less than minimum), invalid. Try $x=476$: $$25,000 \times 476 = 11,900,000$$ Remaining: $$12,062,000 - 11,900,000 = 162,000$$ Try $y=30$: $$4,500 \times 30 = 135,000$$ Remaining after $y$: $$162,000 - 135,000 = 27,000$$ Try $z=0$, $w=11$ (multiple of 10? No, 11 not multiple of 10), invalid. Try $w=10$: $$2,500 \times 10 = 25,000$$ Remaining: $$27,000 - 25,000 = 2,000$$ No SSR points left to cover 2,000. Try $z=0$, $w=11$ invalid. Try $z=0$, $w=12$: $$2,500 \times 12 = 30,000$$ Exceeds 27,000. Try $z=1$ (multiple of 10? No), invalid. Try $z=10$: $$8,750 \times 10 = 87,500$$ Exceeds 27,000. Try $z=0$, $w=10$: Total points: $$11,900,000 + 135,000 + 0 + 25,000 = 12,060,000$$ Short by 2,000. Try $y=31$: $$4,500 \times 31 = 139,500$$ Total: $$11,900,000 + 139,500 + 0 + 25,000 = 12,064,500$$ Exceeds by 2,500. Try $w=9$ (not multiple of 10), invalid. 7. **Conclusion:** This problem requires systematic search or programming to find exact integer solutions respecting all constraints. **Final note:** The problem is a constrained integer linear combination problem best solved with computational tools.