Question

1. a popular kid’s game has 15 tiles and 1 open space. the goal of the game is to rearrange the tiles to put them in order (from least to greatest, starting at the upper left - hand corner and going across each row). use the figure to write the transformation(s) that describe the path of where the 8 tile is currently, and where it must be by the end of the game. can this same translation be used to describe the path of all the tiles?

the figure (a 4x4 grid) has the following tiles:
row 1: 8, 2, 3, 7
row 2: 5, 6, 4, 14
row 3: 1, 9, open space, 13
row 4: 11, 15, 10, 12

Explanation:

First, identify the current position of the 8 tile: in the grid, rows/columns are 1-indexed (top-left to bottom-right). The 8 is at (1,1) (row 1, column 1). The goal position for 8 is (2,4) because the ordered grid is [1,2,3,4;5,6,7,8;9,10,11,12;13,14,15,open], so 8 is in row 2, column 4. The translation from (1,1) to (2,4) is (1,3) (down 1 row, right 3 columns). This translation cannot apply to all tiles because each tile has a unique target position with different row/column shifts.

Answer:

The transformation for the 8 tile is a translation of (1,3) (1 unit down and 3 units right). No, the same translation cannot be used for all tiles.