Prepare for the counterattack

Question

My adviser's brilliance crushed the Blackbeard's fleet, and we held a banquet to celebrate. The banquet was short lived however, for a arrow flew right past me and struck a pillar. Attached was a message from Blackbeard. It reads

Arrr, how dare you attack me branch fleet while I was out swashbucklin'. By the name of meself, I will send you to Davy Jones' locker.

I realized right away that I had made an error in my previous attack. It was indeed too little to be called the Blackbeard Grand Fleet.

Quickly, I dismissed the banquet and summoned my military adviser to discuss how we should prepare. He advised that we strike them in the high sea instead, as fighting at home would potentially put all the citizens in danger. He further said that Blackbeard would most likely sail directly from where the previous engagement was, so we should prepare to scout in that general direction right away.

Thus once again, scouts were sent out. But by the time we received a message, it was near nightfall. It read

We have found a fleet wearing Blackbeard's Jolly Roger. However, because it is dark, I don't see Blackbeard anywhere. To make things worse, the fog makes it hard to pinpoint the location of their ship, so I can only attach a map with how many ship pieces there are on each row and column. Based on my first speculation, there's again 1 flagship, 2 frigates, 3 corvettes, and 4 patrol boats, and I do not believe they touch each other. Not even diagonally.

$$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|l} \hline&&&&&&&&&&4\\ \hline&&&&&\bigcirc&&&&&5\\ \hline&&&&&&&&&&2\\ \hline&&&&&&&&&&2\\ \hline&&&&&&&&&&1\\ \hline&&&&&&&&&&1\\ \hline&&&&&&&&&&1\\ \hline&&&&&&&&&&1\\ \hline&&&&&&&&&&0\\ \hline&&&&&&&&&&3\\ \hline1&1&2&3&1&7&1&2&0&2 \end{array} \quad\quad \bbox[4pt,border: 1px solid black]{ \begin{array}{llc} \bf{Legend:}&\\ \hline \text{Flagship:}& \blacktriangleleft\blacksquare\blacksquare\blacktriangleright \text{or} &\blacktriangle\\ &&\blacksquare\\ &&\blacksquare\\ &&\blacktriangledown\\ \hline \text{Frigate:}& \blacktriangleleft\blacksquare\blacktriangleright \text{or} &\blacktriangle\\ &&\blacksquare\\ &&\blacktriangledown\\ \hline \text{Corvette:}& \blacktriangleleft\blacktriangleright \text{or} &\blacktriangle\\ &&\blacktriangledown\\ \hline \text{Patrol Boat:}& \blacklozenge\\ \hline \text{HUGE rock:}& \bigcirc \end{array}} $$

My adviser looked at the map for a few minutes, and said to me: this is most likely his vanguard fleet; we still need to crush them, but his main fleet should be not far behind, and I should send the scouts there instead. Then, he quickly disappeared into the operation room like before, leaving me pondering how he managed to find the ships using this map.

Can you guys help me figure out the enemy location again?

Answer 1

$$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|l} \hline&&&&&\blacktriangleleft&\blacksquare&\blacktriangleright&&\blacktriangle&4\\ \hline\blacktriangleleft&\blacksquare&\blacksquare&\blacktriangleright&&\bigcirc&&&&\blacktriangledown&5\\ \hline&&&&&\blacktriangle&&\blacklozenge&&&2\\ \hline&&&\blacklozenge&&\blacksquare&&&&&2\\ \hline&&&&&\blacktriangledown&&&&&1\\ \hline&&&\blacklozenge&&&&&&&1\\ \hline&&&&&\blacktriangle&&&&&1\\ \hline&&&&&\blacktriangledown&&&&&1\\ \hline&&&&&&&&&&0\\ \hline&&\blacklozenge&&\blacktriangleleft&\blacktriangleright&&&&&3\\ \hline1&1&2&3&1&7&1&2&0&2 \end{array} $$

First you eliminate the row and column with 0.

$$ \def\b{\color{blue}\blacksquare} \def\ltri{\color{red}\blacktriangleleft} \def\rtri{\color{red}\blacktriangleright} \def\utri{\color{red}\blacktriangle} \def\dtri{\color{red}\blacktriangledown} \def\rsq{\color{red}\blacksquare} \def\rdi{\color{red}\blacklozenge} \begin{array}{|c|c|c|c|c|c|c|c|c|c|l} \hline&&&&&&&&\b&&4\\ \hline&&&&&\bigcirc&&&\b&&5\\ \hline&&&&&&&&\b&&2\\ \hline&&&&&&&&\b&&2\\ \hline&&&&&&&&\b&&1\\ \hline&&&&&&&&\b&&1\\ \hline&&&&&&&&\b&&1\\ \hline&&&&&&&&\b&&1\\ \hline\b&\b&\b&\b&\b&\b&\b&\b&\b&\b&0\\ \hline&&&&&&&&\b&&3\\ \hline1&1&2&3&1&7&1&2&0&2 \end{array} $$

The sixth column can at most have one more free space so we can fill some of it in and add some more free spaces. The free spaces in the middle of columns 5 and 7 are there because any way we could fill the sixth column would block those spaces.

$$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|l} \hline&&&&&?&&&\b&&4\\ \hline&&&&\b&\bigcirc&\b&&\b&&5\\ \hline&&&&&?&&&\b&&2\\ \hline&&&&\b&&\b&&\b&&2\\ \hline&&&&\b&&\b&&\b&&1\\ \hline&&&&\b&&\b&&\b&&1\\ \hline&&&&\b&&\b&&\b&&1\\ \hline\b&\b&\b&\b&\b&?&\b&\b&\b&\b&1\\ \hline\b&\b&\b&\b&\b&\b&\b&\b&\b&\b&0\\ \hline&&&&&?&&&\b&&3\\ \hline1&1&2&3&1&7&1&2&0&2 \end{array} $$

Now we can place the Flagship either in row 2 or in column 6. I excluded row 1 because any way you put it there would block at least two more spaces from row 2 but only one more space is allowed to be free there.

I tried placing the Flagship in the only position of row 2 and the rest solved itself afterwards.

$$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|l} \hline\b&\b&\b&\b&\b&?&&&\b&&4\\ \hline\ltri&\rsq&\rsq&\rtri&\b&\bigcirc&\b&&\b&&5\\ \hline\b&\b&\b&\b&\b&?&&&\b&&2\\ \hline\b&\b&&&\b&&\b&&\b&&2\\ \hline\b&\b&&&\b&&\b&&\b&&1\\ \hline\b&\b&&&\b&&\b&&\b&&1\\ \hline\b&\b&&&\b&&\b&&\b&&1\\ \hline\b&\b&\b&\b&\b&?&\b&\b&\b&\b&1\\ \hline\b&\b&\b&\b&\b&\b&\b&\b&\b&\b&0\\ \hline\b&\b&&&&?&&&\b&&3\\ \hline1&1&2&3&1&7&1&2&0&2 \end{array} $$

step

$$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|ll} \hline\b&\b&\b&\b&\b&\ltri&\rsq&\rtri&\b&\utri&4^*\\ \hline\ltri&\rsq&\rsq&\rtri&\b&\bigcirc&\b&\b&\b&\dtri&5^*\\ \hline\b&\b&\b&\b&\b&?&\b&&\b&\b&2\\ \hline\b&\b&&&\b&&\b&&\b&\b&2\\ \hline\b&\b&&&\b&&\b&&\b&\b&1\\ \hline\b&\b&&&\b&&\b&&\b&\b&1\\ \hline\b&\b&&&\b&&\b&&\b&\b&1\\ \hline\b&\b&\b&\b&\b&?&\b&\b&\b&\b&1\\ \hline\b&\b&\b&\b&\b&\b&\b&\b&\b&\b&0\\ \hline\b&\b&&&?&?&\b&&\b&\b&3\\ \hline1&1&2&3&1^*&7&1&2&0&2^\dagger\\ \end{array} \\ ^*\text{must be filled}\\ ^\dagger\text{at most two} $$

step

$$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|ll} \hline\b&\b&\b&\b&\b&\ltri&\rsq&\rtri&\b&\utri&4\\ \hline\ltri&\rsq&\rsq&\rtri&\b&\bigcirc&\b&\b&\b&\dtri&5\\ \hline\b&\b&\b&\b&\b&?&\b&\rdi&\b&\b&2^*\\ \hline\b&\b&&&\b&?&\b&\b&\b&\b&2\\ \hline\b&\b&&&\b&&\b&\b&\b&\b&1\\ \hline\b&\b&&&\b&&\b&\b&\b&\b&1\\ \hline\b&\b&\b&\b&\b&?&\b&\b&\b&\b&1\\ \hline\b&\b&\b&\b&\b&?&\b&\b&\b&\b&1\\ \hline\b&\b&\b&\b&\b&\b&\b&\b&\b&\b&0\\ \hline\b&\b&&&?&?&\b&\b&\b&\b&3\\ \hline1&1&2&3&1&7^{\dagger\dagger}&1&2^\dagger&0&2\\ \end{array} \\ ^*\text{must be filled}\\ ^\dagger\text{at most two}\\ ^{\dagger\dagger}\text{this column needs a corvette and a frigate in the middle} $$

step

$$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|ll} \hline\b&\b&\b&\b&\b&\ltri&\rsq&\rtri&\b&\utri&4\\ \hline\ltri&\rsq&\rsq&\rtri&\b&\bigcirc&\b&\b&\b&\dtri&5\\ \hline\b&\b&\b&\b&\b&?&\b&\rdi&\b&\b&2\\ \hline\b&\b&\b&&\b&?&\b&\b&\b&\b&2\\ \hline\b&\b&\b&&\b&&\b&\b&\b&\b&1\\ \hline\b&\b&\b&&\b&&\b&\b&\b&\b&1\\ \hline\b&\b&\b&\b&\b&?&\b&\b&\b&\b&1\\ \hline\b&\b&\b&\b&\b&?&\b&\b&\b&\b&1\\ \hline\b&\b&\b&\b&\b&\b&\b&\b&\b&\b&0\\ \hline\b&\b&\rdi&\b&\ltri&\rtri&\b&\b&\b&\b&3^*\\ \hline1&1&2&3&1&7&1&2&0&2\\ \end{array} \\ ^*\text{must be filled but no more frigates}\\ $$

step

$$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|ll} \hline\b&\b&\b&\b&\b&\ltri&\rsq&\rtri&\b&\utri&4\\ \hline\ltri&\rsq&\rsq&\rtri&\b&\bigcirc&\b&\b&\b&\dtri&5\\ \hline\b&\b&\b&\b&\b&?&\b&\rdi&\b&\b&2\\ \hline\b&\b&\b&\rdi&\b&?&\b&\b&\b&\b&2\\ \hline\b&\b&\b&\b&\b&&\b&\b&\b&\b&1\\ \hline\b&\b&\b&\rdi&\b&\b&\b&\b&\b&\b&1\\ \hline\b&\b&\b&\b&\b&?&\b&\b&\b&\b&1\\ \hline\b&\b&\b&\b&\b&?&\b&\b&\b&\b&1\\ \hline\b&\b&\b&\b&\b&\b&\b&\b&\b&\b&0\\ \hline\b&\b&\rdi&\b&\ltri&\rtri&\b&\b&\b&\b&3\\ \hline1&1&2&3^*&1&7&1&2&0&2\\ \end{array} \\ ^*\text{only 2 patrols left for here}\\ $$

last step solving sixth column

$$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|ll} \hline\b&\b&\b&\b&\b&\ltri&\rsq&\rtri&\b&\utri&4\\ \hline\ltri&\rsq&\rsq&\rtri&\b&\bigcirc&\b&\b&\b&\dtri&5\\ \hline\b&\b&\b&\b&\b&\utri&\b&\rdi&\b&\b&2\\ \hline\b&\b&\b&\rdi&\b&\rsq&\b&\b&\b&\b&2\\ \hline\b&\b&\b&\b&\b&\dtri&\b&\b&\b&\b&1\\ \hline\b&\b&\b&\rdi&\b&\b&\b&\b&\b&\b&1\\ \hline\b&\b&\b&\b&\b&\utri&\b&\b&\b&\b&1\\ \hline\b&\b&\b&\b&\b&\dtri&\b&\b&\b&\b&1\\ \hline\b&\b&\b&\b&\b&\b&\b&\b&\b&\b&0\\ \hline\b&\b&\rdi&\b&\ltri&\rtri&\b&\b&\b&\b&3\\ \hline1&1&2&3&1&7&1&2&0&2\\ \end{array} $$