## Ken's POTW

Rows of Checkers
1. Place ten checkers into a 4x4 grid to form the largest number of even rows. An even row has an even number of checkers in it (2 or 4). Rows may be counted vertically, horizontally, or diagonally.
2. Place as many checkers as possible into a 6x6 grid, such that there are no more than two checkers in any straight line, [including all the diagonal and off-diagonal directions.] Two checkers have already been placed in opposite corners, so no more checkers are permitted on that corner-to-corner diagonal.
3. Place sixteen checkers into an 8x8 grid so that there will not be more than two in a row vertically, horizontally, or diagonally [or any direction.] There is one stipulation. Two of the checkers must be placed in the four central squares of the board.

Source: Mathematical Puzzles of Sam Loyd, Volume Two, edited by Martin Gardner, Dover Publications, New York, 1960, #155, #84, #48.

Solutions:
1.  ```. x . x x x . . x x x x . x x .``` Received from Kirk Bresniker, Igor Volkov (and Sam Loyd). There are 16 even rows.
2.  ```x . x . . . . . x . x . x x . . . . . . . . x x . x . x . . . . . x . x``` This is the only solution I've seen so far. Received from Rich Polster, Philippe Fondanaiche Paris France, Igor Volkov (and Sam Loyd). Several people made excellent attempts at this problem but forgot to check the off-diagonal directions (there were several answers with 3 checkers placed in a line of knight's moves, for example.)
Kirk Bresniker, with a computer program, found several other solutions on 10/24/97. He is currently checking for uniqueness and they will be posted soon.
3.  ```. . . . . x . x x . . x . . . . . . . . . x . x . x . x . . . . . . . . x . x . x x . . . . . . . . x . x . . . . . x . . . x .``` This was received from Igor Volkov and is the same solution Sam Loyd gives.
Philippe Fondanaiche, with a computer program, found several other solutions on 10/23/97. He is currently checking for uniqueness and they will be posted soon.

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