Four P Pentominoes (Posted on 2016-03-04)

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	Submitted by Brian Smith
Rating: 4.0000 (1 votes)
Solution:	(Hide)
The 5x5 grid below is marked with four O tiles and 21 X tiles. Any placement of a P pentomino must cover one of these four tiles. (Actually a slightly stronger statment is that any 2x2 square must cover one of these four tiles). XXXXX XOXOX XXXXX XOXOX XXXXX Because there are only four O tiles the maximum number of P pentominoes that can be placed is 4, meaning that it is impossible to place 5 P pentominoes in the 5x5 grid. For any of the other pentominoes to be included with the four P pentominoes that pentomino must be placed solely on the X tiles. The seven grids below show seven different pentominoes A-G each with four P pemtominoes P-S covering a 5x5 square. AAAAA BBBB CCCC D EEE FF G B C D E F GGG DDD E FF G AAAAA BBBBQ CCCCQ DPPPQ PEEEQ PFFQQ PPPQQ PPPQQ BPPQQ PPCQQ DPPQQ PPEQQ PPFQQ PPGQQ PPQQQ PPPQQ PPPQQ DDDQQ PPEQQ PPFFQ RGGGQ RRRSS RRRSS RRRSS RRRSS RRRSS RRRSS RRGSS RRSSS RRSSS RRSSS RRSSS RRSSS RRSSS RRSSS The three pentominoes below cannot be placed without covering an O tile, and therefore cannot cover the 5x5 square along with four P pentominoes. (Again a slightly stronger statement can be made, the skew tetromino cannot be placed without covering one of the O tiles. These three pentominoes and the P pentomino all contain the skew tetromino as a subfigure.) XX XX XXX XX XX XX X X Finally, the U pentomino. Ignoring rotations and reflections there are two distinct placements for the U pentomino: ..... UUU.. U.U.. UPU.. UUU.. PP... ..... PP... ..... ..... In the first figure it is impossible to place a P pentomino in the upper left. The second figure shows how one P pentomino must be placed to avoid having a gap along the left side, but there is still a gap in the bottom that cannot be filled. This shows the U pentomino cannot cover the 5x5 square along with four P pentominoes. In summary, seven of the twelve pentominoes can be combined with 4 P pentominoes to cover a 5x5 square.

	Subject	Author	Date
	Good work	armando	2016-03-05 17:02:41
	re: Sq probe	Steve Herman	2016-03-04 19:17:30
	Sq probe	armando	2016-03-04 14:56:02
	Solutions	Jer	2016-03-04 11:56:12