Smallest Army on a chessboard

This is actually similar question to Biggest army on a chessboard. But it is kinda different way of asking.

In this game, you are supposed to put the least amount of pieces into the board where there will be no left square you can put any pieces. If this question was asked for

• Queen only, the answer will be $5$ as defined in Wolfram.

• Rooks only the answer will be $8$.

• Kings only $9$.

• Bishops only $8$.

• Knights only $12$.

So what if we use Queen, Rooks, Knights, Bishops altogether,

What is the least point you can get with the adjusted penalty points below where there will be no square left on the board you can put any kind of opposite piece? (Notice that no King exists because it makes the question quite easy):

• Queens: 24, Rooks: 15, Knights: 10, Bishops: 15.

These penalties are based off of the other similar question.

• Even with the edit, I don't understand the penalties. Using the other question's system; shouldn't a Queen be 1/5; a Rook 1/8; a King 1/9; etc? – GendoIkari Mar 1 '18 at 21:16
• @GendoIkari it is simply the same formula as previous question, such as 1/8 Bishop or 1/12 for Knight etc, but it is just formed as integers by multiplying them with 120 as you suspected. So the same thing actually. but this time, you want to get lowest point instead of highest point... – Oray Mar 1 '18 at 21:20
• I'm not sure any more about what the question is asking. Are the pieces allowed to attack each other? It seems so from the knights solution. If so, then why would it be impossible to add an extra piece to the board whenever you like? (apart from increasing your penalty). Would the question be better phrased as being about using pieces with the lowest total penalty such that they are attacking all empty squares on the board? – Jaap Scherphuis Mar 1 '18 at 22:46
• @JaapScherphuis When he said you can't put any pieces down, I'm sure he meant that the "army" is all one color, and you couldn't place any piece from the opposing color – ferret Mar 1 '18 at 23:12
• @JaapScherphuis yes, it means no square left where you can put any opposite piece. – Oray Mar 2 '18 at 5:14