King placement problem, brute force version

kingplacement.py

import itertools

# How many kings can be placed on an NxN board such that there are
#  2 kings in each row
#  2 kings in each column?
#  No pair of attacking kings?

def kingPositionsInRow( availableColumns ):
    for k1, k2 in itertools.combinations( availableColumns, 2 ):
            if k2 == k1 + 1:
                continue
            if k2 == k1 - 1:
                continue
            yield (k1, k2)

def discardAttackedColumns( available, k1 ):
    available.discard( k1 - 1 )
    available.discard( k1 )
    available.discard( k1 + 1 )

def updateCounts( prevCounts, k1, k2 ):
    counts = list( prevCounts )
    counts[k1] += 1
    counts[k2] += 1
    return counts

def placeKings( width, remainingRows, prevPlaced, prevCounts,
                fullColumns, availableColumns ):
    if remainingRows == 1:
        for (k1,k2) in kingPositionsInRow( availableColumns ):
            colCounts = updateCounts( prevCounts, k1, k2 )
            if colCounts == [2] * width:
                yield prevPlaced + [(k1,k2)]
    else:
        for (k1,k2) in kingPositionsInRow( availableColumns ):
            #print( "Attempting", (k1,k2), "for row", remainingRows )
            newCount = updateCounts( prevCounts, k1, k2 )
            if newCount[k1] == 2 or newCount[k2] == 2:
                newFull = set( fullColumns )
                if newCount[k1] == 2:
                    newFull.add( k1 )
                if newCount[k2] == 2:
                    newFull.add( k2 )
            else:
                newFull = fullColumns
            nextAvail = set( range( width ) ) - fullColumns
            discardAttackedColumns( nextAvail, k1 )
            discardAttackedColumns( nextAvail, k2 )
            for placement in placeKings( width,
                                         remainingRows - 1,
                                         prevPlaced + [(k1,k2)],
                                         newCount,
                                         newFull,
                                         nextAvail ):
                yield placement

def showPlacements( width ):
    count = 0
    for p in placeKings( width, width, [], [0] * width,
                         set(), set( range( width ) ) ):
        count += 1
        print( p )
    print( "total", count )

def countPlacements( width ):
    count = 0
    example = None
    for p in placeKings( width, width, [], [0] * width,
                         set(), set( range( width ) ) ):
        example = p
        count += 1
    print( "total", count )
    print( "example", example )
    
def showExample( kings ):
    width = max( max(k1,k2) for (k1,k2)  in kings ) + 1
    for k1,k2 in kings:
        output = ["."] * width
        output[k1] = "K"
        output[k2] = "K"
        print( "".join( output ) )