Let's play mini-Sudoku!
We wish to place an "X" in four cells, such that there is exactly one "X" in each row, column, and 2x2 outlined box. For example:
In how many ways can we do this?
Terminology: A cell is one of small 1x1 spaces. A box is one of four outlined 2x2 group of cells. Please use this terminology so there is not ambiguity between you and your audience.
+26367 Let's play mini-Sudoku!
We wish to place an "X" in four cells, such that there is exactly one "X" in each row, column, and 2x2 outlined box. For example:
In how many ways can we do this?
Each cell has a number:
\(\begin{array}{|r|r|r|r|} \hline I_1 & I_2 & II_1 & II_2 \\ \hline I_3 & I_4 & II_3 & II_4 \\ \hline III_1 & III_2 & IV_1 & IV_2 \\ \hline III_3 & III_4 & IV_3 & IV_4 \\ \hline \end{array}\)
Then we have 16 varying mini-Sudokus.
They are:
\(\begin{array}{|rrrrr|} \hline 1 & I_1 & II_3 & III_2 & IV_4 \\ 2 & I_1 & II_3 & III_4 & IV_2 \\ 3 & I_1 & II_4 & III_2 & IV_3 \\ 4 & I_1 & II_4 & III_4 & IV_1 \\ \hline 5 & I_2 & II_3 & III_1 & IV_4 \\ 6 & I_2 & II_3 & III_3 & IV_2 \\ 7 & I_2 & II_4 & III_1 & IV_3 \\ 8 & I_2 & II_4 & III_3 & IV_1 \\ \hline 9 & I_3 & II_1 & III_2 & IV_4 \\ 10 & I_3 & II_1 & III_4 & IV_2 \\ 11 & I_3 & II_2 & III_2 & IV_3 \\ 12 & I_3 & II_2 & III_4 & IV_1 \\ \hline 13 & I_4 & II_1 & III_1 & IV_4 \\ 14 & I_4 & II_1 & III_3 & IV_2 \\ 15 & I_4 & II_2 & III_1 & IV_3 \\ 16 & I_4 & II_2 & III_3 & IV_1 \\ \hline \end{array}\)
+128474
Here's a slightly different approach :
We have 4 ways to choose the placement of the X in the first row
Then, we have 2 ways to choose the placement of the X in the second row.....choose either column in the box not occupied by the first X
Then we have 2 ways to choose the placement of the X in the third row......choose either column not occupied by the first two X's
And we only have 1 remaining choice of boxes in the last row
So...
4 * 2 * 2 * 1 = 16 ways
