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# Square Free

## Question

Remove 4 pieces from the above position.. then place them back on any free squares of the chessboard  so that no square can be formed by any four pieces on corners of a square .  Which pieces to choose and where to position them?

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This is base on my posted question : " Anti Square" .About most number of pieces on chess board without forming a square. Yet 34 may not be the maximum.

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I'll stand with the three solutions found.

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Edited by rocdocmac
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Clarification:

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Do you agree the pieces at { e1 c3 g3 e5 } form a square?

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16 hours ago, rocdocmac said:
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I have identified 7 squares on the board as it is currently. Correct? WrongI now have found 8

There are 19 pieces involved in these 7 squares. Correct? Wrong20 pieces involved.

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I believe the 8 squares you've identified are the following:
{a1, c1, c3, a3},
{a1, e1, e5, a5},
{c1, h1, h6, c6},
{a3, d3, d6, a6},
{c3, h3, h8, c8},
{d3, g3, g6, d6},
{c3, d1, f2, e4},
{c3, e1, g3, e5}
If this is not correct, there are more than 8 squares.

16 hours ago, rocdocmac said:
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I have identified 7 squares on the board as it is currently. Correct? WrongI now have found 8

There are 19 pieces involved in these 7 squares. Correct? Wrong20 pieces involved.

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The count of 20 pieces involved is wrong as there are 21 pieces involved in total in corners of these 8 squares:
{a1, c1, d1, e1, h1, f2,  a3, c3, d3, h3, g3, e4, a5,  e5, a6, c6,  d6, g6, h6, c8, h8}

.

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36 minutes ago, plasmid said:
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?

square formed 45 degrees from orthogonal using the black king?

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2 hours ago, bonanova said:

Clarification:

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Do you agree the pieces at { e1 c3 g3 e5 } form a square?

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Yes of course..  consider the upright and tilted squares.

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Edited by TimeSpaceLightForce
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Silly question ... why an extra pawn on each advancing side (18 chess pieces in total) or is that just part of the puzzle to work?

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The added black & white pawns make the total pieces 34 (discovered max ). Note they are but markers.

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Dumb question from me! ... I suppose one just has to replace all the pieces with "counters" and follow the rule!

"We still need a lot of education" a la Pink Floyd!

Edited by rocdocmac
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Hint: Only remove the square corner pieces

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22 hours ago, TimeSpaceLightForce said:
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Hint: Only remove the square corner pieces

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I have identified 7 squares on the board as it is currently. Correct?

There are 19 pieces involved in these 7 squares. Correct?

If you say that only the square corner pieces should be removed, does that mean pieces from only one square?

I think i have identified the four pieces. I can place three of them back without making extra squares, but I am battling placing the last one back! Think I'll go through my process again.

Edited by rocdocmac
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I have identified 7 squares on the board as it is currently. Correct? WrongI now have found 8

There are 19 pieces involved in these 7 squares. Correct? Wrong20 pieces involved.

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Latest count: 9 squares, 24 pieces involved (8 of these 24 appear as part of two or more squares).

Edited by rocdocmac
wording
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Edit:  23 pieces involved in 9 squares (8 pieces fit into more than one square).

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34 minutes ago, rocdocmac said:

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Edit:  23 (wrong) pieces involved in 9 squares (8 pieces fit into more than one square). Recounted 24 pieces.

Edited by rocdocmac
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Is there more than one solution?

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Looks like there are even more perfect squares on the original layout!

My latest count is 11 squares with 27 pieces being involved and 11 of them found in more than one square.

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I myself did not own the 34 pieces solution. Someones code did it; like placing 3 marks randomly.. then one by one marking every square and  Tested if  it  is 4th corner or not..My original manual solution is just 31 pieces thats why I posted before using a chess set 32 pieces

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There are  more  than 300 possible squares  on a chessboard.. i guess

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3 hours ago, rocdocmac said:
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Looks like there are even more perfect squares on the original layout!

My latest count is 11 squares with 27 pieces being involved and 11 of them found in more than one square.

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Edited by TimeSpaceLightForce
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3 hours ago, rocdocmac said:
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Looks like there are even more perfect squares on the original layout!

My latest count is 11 squares with 27 pieces being involved and 11 of them found in more than one square.

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My own count has also increased, both with squares and pieces.  12 squares and 28 pieces. The four additional squares are: {a6, c3, f5, d8}, {e5, d3, f2, g4}, {b2, c1, d2, c3}, {a5, d3, f6, c8} -- the previously listed being: {a1, c1, c3, a3},
{a1, e1, e5, a5}, {c1, h1, h6, c6}, {a3, d3, d6, a6}, {c3, h3, h8, c8}, {d3, g3, g6, d6}, {c3, d1, f2, e4}, {c3, e1, g3, e5}

The twenty-eight pieces are those at {a1, a3, a5, a6, b2, c1, c3, c6, c8, d1, d2, d3, d5, d6, d8, e1, e4, e5, f2, f5, f6, g3, g4, g6, h1, h3, h6, h8}.
Removing the two pieces at c3 and d3 will leave of these twelve squares but the two {a1, e1, e5, a5} and {c1, h1, h6, c6}. From each of these remaining two one of the four corners of each will also need to be removed.

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I shall stand with a total of 12 squares since I've also found that figure later this afternoon, but I only found 27 pieces involved (12 being part of one or more squares). I still have to compare your 28 pieces with my 27 to see the discrepancy!

Edited by rocdocmac
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1 hour ago, DejMar said:

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The twenty-eight pieces are those at {a1, a3, a5, a6, b2, c1, c3, c6, c8, d1, d2, d3, d5, d6, d8, e1, e4, e5, f2, f5, f6, g3, g4, g6, h1, h3, h6, h8}.

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There is no d5. Check again ... it should be 27 pieces!

Edited by rocdocmac
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A possible solution:

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Remove the four pieces {a5, c3, d3, c6} and replace them at {a4, e8,  g7, g2}.

4 hours ago, rocdocmac said:

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Correct, no d5. 27 pieces.

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10 hours ago, DejMar said:

Remove the four pieces {a5, c3, d3, c6} and replace them at {a4, e8,  g7, g2}.

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Invalid solution ... a 5x5 square is formed by {b2, b7, g7, g2}

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?

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I believe that this is one possible solution ...

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Edited by rocdocmac
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