How to place 2 bishops and 7 knights on a 4x4 chessboard without any of them attacking other pieces?
There are no other restrictions.
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is this possible– cmgchessMar 16, 2022 at 16:11
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@cmgchess I was told that one student at my college solved it before7 years, but I doubt it.– Josip MaričevićMar 16, 2022 at 16:33
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i found this geeksforgeeks.org/… and from this can get board positons for 4*4 with 7 knights. see if you can more than 1 unattacked square for bishop– cmgchessMar 16, 2022 at 16:35
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1 Answer
This is not possible.
Proof: Consider all the possible configurations where there are 8 knights and 0 bishops and then see if it is possible to remove one knight and place two bishops on empty, unattacked squares.
In the top left diagram only the knights on b1, b4, c1 and c4 can be replaced with bishops. But although you can replace one of these knights with a bishop you cannot place another bishop on the board because all the empty squares are attacked. Hence if you place 7 knights on the board then you can place at most one bishop. You can replace two knights with bishops but then you have only 6 knights and no more knights can be placed.
Note that only with the top left and the rotated equivalent at the middle bottom can replace a knight with a bishop. In all the other cases replacing a knight with a bishop will leave a knight attacked.
For this I found out possible combinations to have 8 unattacked knights and there were 6 (less than 6 if symmetry considered). Notice how each unoccupied square is attacked by 2 knights. So you are only free to remove 1 knight to occupy a bishop. But then you have no squares left for a 2nd bishop
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1Why do you have to start with 8 unattacked knights? Why can't you start with 8 knights placed so that two of them attack each other? Of course you have to remove one of those two knights before looking for squares for the bishops.– bofMar 17, 2022 at 1:56
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Maybe it would be easier to consider all ways of placing 2 nonattacking bishops, and observe that in each case the maximum number of knights that can be added is no more than 6. If I counted right, there are (considering symmetry) just 21 ways to place two bishops, and just 13 ways such that they don't attack each other.– bofMar 17, 2022 at 2:00
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@bof i used a code snippet to generate all positions with 8 unattakced knights. after that only i tried the rest– cmgchessMar 17, 2022 at 8:41
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1How do you know you can start with 8 unattacked knights? Are you sure that every configuration of 7 unattacked knights can be obtained from one with 8 unattacked knights?– bofMar 17, 2022 at 11:45