Color sudoku puzzle

Probability Level 4

How many ways of coloring the next square with four colors are so that:

$1)$ Each row has the four colors

$2)$ Each column has the four colors

$3)$ Each column square of $2\times 2$ with bold edge has the four colors

1 solution

Victor Paes Plinio
Sep 2, 2015

Look this square above. We have four areas A, B, C and D, and each area have four little squares.

Let's change the colors by numbers (traditional sudoku)

To fill the area A with the numbers 1, 2, 3 and 4 in crescent order, we have the following situation:

We can put the number 1 in any of the 4 little squares;
When we putted the number 1, there are left three little squares to choose for number 2;
When we putted the numbers 1 and 2, there are left two little squares to choose for number 3; -Finally we will have only one little square for the number 4.

In total we have $4!=4\times3\times2\times1=24$ ways to fill the area A.

Let's fill the area D.

-To put the number 1, we have four little squares.

We have now 3 ways to fill the area D.

in the image above we can see these three ways. The gray square no works because we will have problem to put fill the other areas.

The red marked square we don't can fill.

Thus we will have $4\times3=12$ for area D.

When the areas A and D are filled, we will only have one way to fill the area B and C.

Thus the total number of squares is $24\times12=288$