You have an empty 8 × 8 chessboard several pawns of the same color.
What is the maximum number of pawns you can put on the chessboard without the pawns defending/attacking each other?
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That has to be the correct answer. The question states that we have "several pawns if the same colour", pawns of the same colour cannot take one another so there is nothing stopping the entire board being filled with them.
yea i agree
Relevant wiki: Pigeonhole Principle
Consider breaking the 8 × 8 chessboard into sixteen 2 × 2 squares.
In each of this
2
×
2
, it is evident that we can't fill in all 4 pawns. Otherwise there will be pawns attacking each other.
Similarly, it is also true that we can't fit in 3 pawns in otherwise there will be one pawn attacking another pawn.
This leaves us with 2 pawns: We arrange these 2 pawns in such a way that only 2 they are in the same row or the same column so that they are not attacking each other.
Hence, we can fill in a maximum of 2 pawns for each 2 × 2 squares. And so, we can fill in a maximum of 1 6 × 2 = 3 2 pawns in a 8 × 8 chessboard. The only thing left to do is to show that it is possible. A possible configuration is to place all the pawns in the rows 1, 3, 5 and 7. And we're done.
Great solution!
I hope you will not mind if I remark that you should also say how to place this pawns in each of these 2X2 squares such that they do not attack each other such that you check if it is possible in principle. By this understanding of the way the configuration of pawns of some 2X2 squares determine the possible configurations of pawns in the other 2X2 squares you will be able to understand completely what and how the configuration of pawns can look generally though this might just be a little bit more complicated to do terms of 2X2 squares where they do not behave the same anyways.
They could only attack each other if they were of differing colours. The question states that you have "several pawns of the same color", so the entire board can be filled with pawns.
Relevant wiki: Chess Puzzles
NOTE: There are lots of ways to get to the same solution, but I will just look at one.
Let's take a look at the movement of a pawn. A pawn only moves forward and captures diagonally . Therefore, any pawn cannot be put diagonally next to another pawn since that pawn would attack it.
We can put a whole row of pawns on the first rank:
but now all the pawns
control
the squares in front of it:
so we cannot put any pawns on the 2nd rank so we'll have to skip a rank and place a row of pawns on the 3rd.
Now we'll have to skip a row and place pawns on the 5th rank (because of similar reasons above)
We'll have to skip the 6th rank as well and place pawns on the 7th rank.
We cannot place any pawns on the 8th rank because of similar reasons stated above
Now if we count the amount of pawns on the board, we will see that we have 32 pawns . The problem asked for the maximum percent of pawns you can put on the chessboard. The percent is calculated by: 3 1 5 3 2 × 1 0 0 ≈ 1 0 . 1 6 which is the answer.
You haven't proved that there is not another configuration that let us place more than 32 pawns satisfying the constraints. You have only proved that there exists at least one configuration that let us place 32 pawns... hint: Pingeonhole principle.
All the pawns are the some colour, so there is nothing to stop every square being filled (pawns can only attack the opposite colour). The answer is 64.
But if all the pawns have the same colour then they can't attack each other, right?
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I think the answer should be 64 because pawns of the same colour can't attack each other.