# Problem #9

You are given four transformations of a 3×3 grid:

• A takes the 2×2 square in the upper left-hand corner and rotates it 90 degrees clockwise.
• B takes the 2×2 square in the upper right-hand corner and rotates it 90 degrees clockwise.
• C takes the 2×2 square in the lower left-hand corner and rotates it 90 degrees clockwise.
• D takes the 2×2 square in the lower right-hand corner and rotates it 90 degrees clockwise.
For example the result of performing transformation A followed by transformation D is shown below.
• Show that begining with the grid numbered as above, any permutation of the numbers can be realized by a finite sequence of A's, B's, C's and D's.

• Find the shortest possible sequence of such transformations that will switch the positions of "1" and "9" while leaving the remaining numbers fixed.

• What is the least number n such that any permutation can be realized by a sequence of n or fewer transformations?

Back to the Archives

Back to the Math Department Homepage.