Question 5
The game Arrasta Um uses a square grid and round pieces, one black and the rest white. Put one piece in each cell except one, which is left empty. A move slides a piece from an adjacent cell into the empty cell. The game ends when the black piece reaches the top right corner. Here is an example finishing a $2\times2$ board in four moves.
Initial position
This sequence is described by $(\uparrow,\leftarrow,\downarrow,\rightarrow)$.
a) Describe how to finish in six moves on the following $3\times3$ board.
b) Describe how to finish in ten moves on the following $3\times3$ board.
c) Show that on an $n\times n$ board as shown, it is possible to finish in $6n-8$ moves.
Initial position
This sequence is described by $(\uparrow,\leftarrow,\downarrow,\rightarrow)$.
a) Describe how to finish in six moves on the following $3\times3$ board.
b) Describe how to finish in ten moves on the following $3\times3$ board.
c) Show that on an $n\times n$ board as shown, it is possible to finish in $6n-8$ moves.