MathDB

The following picture illustrates the model of the Tháp Rùa (The Central Tower in Hanoi), which consists of

3

levels. For the first and second levels, each has

10

doorways among which

3

doorways are located at the front,

3

at the back,

2

on the right side and

2

on the left side. The third level is on the top of the tower model and has no doorways. The front of the tower model is signified by a circle symbol on the top level (Figure). We paint the tower model with three colors: Blue, Yellow and Brown by fulfilling the following requirements: (a) The top level is painted with only one color. (b) The

3

doorways at the front on the second level are painted with the same color. (c) The

3

doorways at the front on the first level are painted with the same color. (d) Each of the remaining

14

doorways is painted with one of the three colors in such a way that any two adjacent doorways with a common side on the same level, including the pairs at the same corners, are painted with different colors. How many ways are there to paint the first level? How many ways are there to paint the entire tower model? https://cdn.artofproblemsolving.com/attachments/f/9/2249f8595a8efe711680f3dfb8ff959c140a21.png

Problem Statement