# Black War Heroes or Whites Save the Queens?

After Black Cavalry successfully captured White princesses,

### White empire learns their Black opponent is hungry for more and wants White Twin Queens dead!

White will never let such a disaster happen and prepare their defense.

### White castle is built by white's women and men's hands. A billion endeavor for a single white case. A billion ingot for a single black case. Most tremendous labor in White's empire history resulting in 80 more cases.




### White's empire also made the greatest academic investment, to bear the smartest researchers in all fields making ALL white Pawns and both Queens more powerful.

Now, each white pawns can kill an opponent on both top diagonals, as they always did, plus on both down diagonals making them 4 cases to threat. That's not all. Each white pawn is now equipped with a poison blade which revenge him and poisons his murderer and making him impossible to perform any move or attack for the next turn.

Ladies Queens have a special tunnel within the castle making them return to their initial case instantly after killing a Black piece.




### Now, I must warn you. As your black right hand, the following is top secret.

Our best spies got white's castle's exact map to prepare our battle.

Map Top Right

Map Top Left

# How our army attacks and would win

Do not hesitate to look for the small example at the puzzle's end.

## 1. Battle's length

We will only have $$20$$ moves before White's empire decide to make their alive queens escape. If exceeding 20 moves, battle is lost.

## 2. Army's cost

On each move we can place black knights or black bishops. All other pieces, towers, pawns, etc. stay in our castle to avoid counter attack. A black attacking piece can be placed on the check board on a case if and only if it is empty. At move $$i\in N=\{1,2,3,...,20\}$$ placing one piece costs $$i$$ units of gold. Example : if we decide to place 2 knights and 3 bishops on step $$1$$, that costs $$5$$ gold. 3 knigths on step 3 costs $$9$$ gold.

## 3. Army's skills

For each $$i\in N$$, we have to decide how many knights and bishops we will send on the battle field. After placing them, each black piece either moves or attacks any piece we choose.

## 4. White defense's skills

Between each steps $$i$$, i.e. at most $$19$$ times, Whites defend according to the following rules:

1. Black pieces closest to queens killed first
2. Kill with white king first - note : killing a King isn't more important than killing any white piece except for twin queens, unlike classic chess
3. Kill with white towers on second
4. Kill with pawns on third
5. Then kill with bishops
6. After that, kill with cavalry
7. Finally twin Queens attack if necessary, then return to their origin case.

## 5. Battle's victory

Battle is won when both white queens are killed.

Killing first queen at step $$i\in N$$ rewards $$2000-100i$$ gold. Killing last queen at step $$i\in N$$ rewards $$10~000 - 100i$$ gold.

# Exemple of battle

Here is a small example on a $$3\times 3$$ check-board with only one queen so you can understand the rules' spirit.

There will only be one Queen for this small example and battle is won when this White queen dies. First step costs $$2$$ golds for Blacks. One bishop and one knight.

Black attacks

White defends according to white defense rules on paragraph 4. White defense's skills.

Step 2. costs 6 more golds for black empire. Black decides to move no piece.

White's defense

Step 3. Final one because costs 3 golds and Blacks win

On this example, Black win $$2~000 - 30\times 3 - 11 = 1~889$$ units of gold but you can easily see this is not an optimal solution!

# Surprise on battle day

This section is made thanks to @Don Thousand's and AxiomaticSystem's correct answers. That's why I like Puzzling.SE, you can be so easily surprised by the efficiency of the community for any puzzle you'll create!

One hour before the battle day, black empire discovers every empty case on the battle field that was empty in the map we knew has a pawn in it, except all cases on the borders.

Moreover, White scientists equipped all units with anti cavalry armors making them attacking only starting from step 4.

Finally, Whites got tactical advantage by attacking after Blacks placing pieces, and after black attacks. Instead of only after black attacks in our plans...

## ...Black War Heroes or Whites Save the Queens?

• This puzzle is not well defined, since it doesn't resolve the order of capturing for pieces of the same type. This actually has an impact on potential solutions. – Don Thousand May 15 at 12:43
• @DonThousand, this is voluntary. When there is any draw between two pieces that can be captured, you decide which one you want to be captured. Actually you will probably decide in a way that helps you improve your solution. – JKHA May 15 at 12:50
• Perhaps a variant where there are fewer pieces but we can assume best play by White would be a good follow-up... – AxiomaticSystem May 15 at 13:34
• @JKHA I'll be honest. Your update to the problem isn't very interesting. The solution is in $3$ moves, regardless of whether White's pieces are attacking or not. – Don Thousand May 15 at 13:49
• @DonThousand, I've made an update after the update. Then I'll stop, you guys are too good at it x') – JKHA May 15 at 13:50

This answer seems too simple to be correct, so I'm probably misunderstanding some of the rules, but I claim that I can get

11594 gold

as follows:

Black Turn 1: Place bishops on d4 and i9, and attack the rooks on e5 and h8
White Turn 1: e5 and h8 are only protected by rooks and queens, so rooks will defend. The bishop on e5 is captured by a rook from either f5 or e6 and the bishop on h8 is captured by a rook from either g8 or h7.
Black Turn 2: Place knights on the two squares that were just vacated by White's rooks. These attack the queens for victory. (f5/e6 threaten g7 and g8/h7 threaten f6)
Costs: Killing both queens nets 12000 gold. Killing them on step two loses me 400 gold, and placing two units on each of the first two turns costs another 6.

Is this optimal?

With this understanding of the rules, yes. Clearly, there is no way to kill either queen on turn 1, and both sacrifices are required in order to get close enough.

A variation, to distinguish from the other answer:

Bd4xe5 can be replaced with either Nc8xe7 or Nh3xg5, and likewise Bi9xh8 with either Nj5xh6 or Ne10xf8.

EDIT: Ah! A surprise! How will I get through these pawns?

Black 1: Na7xc8 Nl6xj5 (these knights are paralyzed, but they die anyway so I don't care)
White 1: Since I get to resolve the pawn conflicts, I choose d7xc8 i6xj5
Black 2: Nd7xf6 Ni6xg7 for the win, with same rewards as before!
And if knights are forbidden I'll need three turns (11388 gold):
Black 1: Bl3xk4 Ba10xb9
White 1: j5xk4 c8xb9
Black 2: Bj5xi6 Bc8xd7
White 2: Rh6xi6 Re7xd7 (Rooks have priority over pawns)
Black 3: Bh6xg7 Be7xf6

EDIT FURTHER: Oh dear...
With White interrupting, I'm going to need

11 steps!
What follows will describe half the action, the action on the other side will be a 180-degree rotation.
Turn 1: Place bishops on a1, a5, e1, f1, h1. White counterattacks with b4xa5, d2xe1, e2xf1, g2xhi. Bishop on a1 attacks the rook on b2, and White recaptures with a3xb2. (5 gold)
Turn 2: Place bishops on a3, e2, g2. White counterattacks with b2xa3, d3xe2, f3xg2. (11 gold)
Turn 3: Place bishops on a1 and f3. White counterattacks with Rf4xf3. Bishop on a1 attacks the knight at c3, and White recaptures with d4xc3. (17 gold)
Turns 4-8: Place a bishop on a1 and capture the piece at c3. White recaptures with the four knights on Turns 4-7 and has no valid response on Turn 8. (47 gold)
Turn 9: The bishop on c3 attacks the rook at e5. White recaptures with Re6xe5. (47 gold)
Turn 10: Place bishops on d5 and e6. White counterattacks with Re5xd5 and Kf7xe6. (67 gold)
Turn 11: Ne4xf6++ (78 gold)
Final cost: 12000 for two kills - 2200 for 11 turns each - 156 for two armies of 78 gold each = 9644 gold.

• Very nice ;) I didn't see your Don Thousand's variation when adding my surprise, please check out the map :) – JKHA May 15 at 13:34
• Already responded to the surprise :) – AxiomaticSystem May 15 at 13:36
• Are we like, making a real battle, surprise by surprise and solutions by solutions haha. I want to give you many +1s ^^' Check out, I hope so, final one! If you do crack it instantly, I'm stopping here, and you get final selected answer. You solve my whole puzzle so quickly haha congrats, that proves your efficency – JKHA May 15 at 13:42
• Finally, Whites got tactical advantage by attacking after Blacks placing pieces, and after black attacks. Instead of only after black attacks in our plans. I think I've edited too slowly – JKHA May 15 at 14:01
• I'm working on a strategy for the latest version... Pretty tricky! – AxiomaticSystem May 15 at 14:05

# Blitzkrieg

Surprisingly, this attack can be accomplished in just $$2$$ moves!

Move $$1$$:

Note that this diagram only includes the top-right part of the castle, but there is a mirror image bishop on the other side of the rook barrier. Both bishops capture rooks.

Move $$2$$:

Since one of the rooks must capture the bishop since the king can't, the vacancy left by the rook can let knights directly attack the queen and finish the job.

So, our final earnings are $$2000+10000-200-200-2-4=11594$$

It's easy to see that this attack can't be done in $$1$$ move, so the only way that a solution can beat this is if it can finish the attack in $$2$$ moves with a cost of less than $$6$$. I don't have a proof for why this isn't possible but it seems pretty unlikely.

• Agh, didn't even get to see this! – AxiomaticSystem May 15 at 13:03
• I love your answer. Didn't see it at all! There is a new section on the puzzle that his dedicated to you. I will select your answer only if no one can prove an optimal on the puzzle's update. You get +1, that's sure – JKHA May 15 at 13:09