+614 
Using only the paths and directions shown, how many different routes are there from M to N?
An equilateral triangle is originally painted black. Each time the triangle is changed, the middle fourth of each black triangle turns white. After five changes, what fractional part of the original area of the black triangle remains black?
Each of the three large squares shown below is the same size. Segments that intersect the sides of the squares intersect at the midpoints of the sides. How do the shaded areas of the squares compare?
+12528 4 ways
Change1: 3/4
Change2: 9/16
Change3: 27/64
Change4: 81/256
Change5/ 243/1024
...............................
Change n: 3^n/4^n
(B)
+614 Thank you; but for the first one i got 6. Can you tell me how you got 4?
+26367 homework questions
Using only the paths and directions shown, how many different routes are there from M to N?
Adjacency matrix directed: Source: https://en.wikipedia.org/wiki/Adjacency_matrix
If A is the adjacency matrix of the directed or undirected graph G,
then the matrix A^n (i.e., the matrix product of n copies of A) has an interesting interpretation:
the element (i, j) gives the number of (directed or undirected) walks of length n from vertex i to vertex j.
\(A = \begin{array}{|c|c|c|c|c|c|c|} \hline & M & A & B & D & C & N \\ \hline M & 0 & 1 & 1 & 0 & 0& \color{red}0 \\ \hline A & 0 & 0 & 0 & 1 & 1& 0 \\ \hline B & 0 & 1 & 0 & 0 & 1& 1 \\ \hline D & 0 & 0 & 0 & 0 & 1& 0 \\ \hline C & 0 & 0 & 0 & 0 & 0& 1 \\ \hline N & 0 & 0 & 0 & 0 & 0& 0 \\ \hline \end{array} ~\text{There is $0\times$ 1-way route }\)
\(A^2 = \begin{array}{|c|c|c|c|c|c|c|} \hline & M & A & B & D & C & N \\ \hline M & 0 & 1 & 0 & 1 & 2 & \color{red}1 \\ \hline A & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \hline B & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \hline D & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \hline C & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \hline N & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \hline \end{array} ~\text{There is $1\times$ 2-way route }\)
\(A^3 = \begin{array}{|c|c|c|c|c|c|c|} \hline & M & A & B & D & C & N \\ \hline M & 0 & 0 & 0 & 1 & 2 & \color{red}2 \\ \hline A & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \hline B & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \hline D & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \hline C & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \hline N & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \hline \end{array} ~\text{There are $2\times$ 3-way routes }\)
\(A^4 = \begin{array}{|c|c|c|c|c|c|c|} \hline & M & A & B & D & C & N \\ \hline M & 0 & 0 & 0 & 0 & 1 & \color{red}2 \\ \hline A & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \hline B & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \hline D & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \hline C & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \hline N & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \hline \end{array} ~\text{There are $2\times$ 4-way routes }\)
\(A^5 = \begin{array}{|c|c|c|c|c|c|c|} \hline & M & A & B & D & C & N \\ \hline M & 0 & 0 & 0 & 0 & 0 & \color{red}1 \\ \hline A & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \hline B & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \hline D & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \hline C & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \hline N & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \hline \end{array} ~\text{There is $1\times$ 5-way route }\)
\(A^6 = \begin{array}{|c|c|c|c|c|c|c|} \hline & M & A & B & D & C & N \\ \hline M & 0 & 0 & 0 & 0 & 0 & \color{red}0 \\ \hline A & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \hline B & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \hline D & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \hline C & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \hline N & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \hline \end{array} ~\text{There is $0\times$ 6-way routes }\)
1+2+2+1 = 6 different routes are there from M to N
