A-7 Four pentominoes can be put together to make a The Duplication Problem. copy of the pentomino which is two times as wide and two times as high as P the original piece. Make a duplication of any pentomino other than the Two of them cannot be P. done. Which ones? Notice that the solution of the duplication above in fact uses its smaller P counterpart in the solution. Find a solution of the duplication that does not P use the pentomino. For every other duplication, determine if there is a P solution that does use its smaller counterpart and then determine if there is a solution that does not use its smaller counterpart. Problems A-8 through A-11: Simultaneous Solutions. A pattern is and you need to use some of the pentominoes to cover the pattern. You then need to use some of the remaining pentominoes to cover the same pattern. This process is called finding of the given simultaneous solutions pattern. A-8 Find three simultaneous solutions to each of the 10-square patterns below. (a) (b) (c) (d) A-9 Find simultaneous solutions to each of the 15-square patterns below. (a) (b) A-10 Determine how many pairs of rectangles can be Simultaneous Rectangles. constructed simultaneously using a single set of pentominoes. As an example, problem A-1 demonstrates that two 3x5 rectangles can be constructed simultaneously. A-11 Find all pairs of pentominoes such that their Simultaneous Duplications. duplications can be constructed simultaneously. A-12 The image below shows the full set of twelve A Pentomino Farm. pentominoes arranged as a to enclose a The rule used to join them fence field. is that they must touch along the full edge of a square and not just at the
A-7 The Duplication Problem. Four pentominoes can be put together to make a copy of the P pentomino which is two times as wide and two times as high as the original piece. Make a duplication of any pentomino other than the P. Two of them cannot be done. Which ones? Notice that the solution of the P duplication above in fact uses its smaller counterpart in the solution. Find a solution of the P duplication that does not use the P pentomino. For every other duplication, determine if there is a solution that does use its smaller counterpart and then determine if there is a solution that does not use its smaller counterpart. Problems A-8 through A-11: Simultaneous Solutions. A pattern is and you need to use some of the pentominoes to cover the pattern. You then need to use some of the remaining pentominoes to cover the same pattern. This process is called finding simultaneous solutions of the given pattern. A-8 Find three simultaneous solutions to each of the 10-square patterns below. (a) (b) (c) (d) A-9 Find simultaneous solutions to each of the 15-square patterns below. (a) (b) A-10 Simultaneous Rectangles. Determine how many pairs of rectangles can be constructed simultaneously using a single set of pentominoes. As an example, problem A-1 demonstrates that two 3x5 rectangles can be constructed simultaneously. A-11 Simultaneous Duplications. Find all pairs of pentominoes such that their duplications can be constructed simultaneously. A-12 A Pentomino Farm. The image below shows the full set of twelve pentominoes arranged as a fence to enclose a field. The rule used to join them is that they must touch along the full edge of a square and not just at the
pentominos-2017.docx