# How many positions can the queen control? [closed]

Given the size of board (n) and the position of the queen (r,c) how can we calculate the total number of squares which can be controlled by the queen?

## closed as off-topic by elias, Glorfindel, Mithrandir, Beastly Gerbil, Rubio♦Jan 29 '17 at 22:23

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• "This question is off-topic as it appears to be a mathematics problem, as opposed to a mathematical puzzle. For more info, see "Are math-textbook-style problems on topic?" on meta." – elias, Glorfindel, Beastly Gerbil, Rubio
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• You can count them! :D – boboquack Jan 29 '17 at 8:29

## 1 Answer

Assuming the queen controls its own square and 0-based indexing, we have:

• $1$ - its own square
• $2n-2$ - the rows and columns without its own square
• $n-1-|r+c-n+1|$ - the number of squares on one diagonal
• $n-1-|r-c|$ - the number of squares on the other diagonal

Add them all up and we have:

$$4n-3+|r+c-n+1|+|r-c|$$