There is a solid surrounded with four equilateral triangles as shown in a figure.
The point P is at the vertex A at first and moves to one of other three vertex for every second.
For example, as for ways of movement where the point P is at the vertex A in 2 seconds, there is A→B→A, A→C→A and A→D→A.
(1) In accordance with the example, write ways of movement altogether where the point P is at the vertex A in 3 seconds.
Moreover, how many ways of movement where the point P is at the vertex B, C, and D in 3 seconds, respectively?
(2) How many ways of movement where the point P is at the vertex A in 4 seconds?
(3) How many ways of movement where the point P is at the vertex A in 5 seconds?
The point P is at the vertex A at first and moves to one of other three vertex for every second.
For example, as for ways of movement where the point P is at the vertex A in 2 seconds, there is A→B→A, A→C→A and A→D→A.
(1) In accordance with the example, write ways of movement altogether where the point P is at the vertex A in 3 seconds.
Moreover, how many ways of movement where the point P is at the vertex B, C, and D in 3 seconds, respectively?
(2) How many ways of movement where the point P is at the vertex A in 4 seconds?
(3) How many ways of movement where the point P is at the vertex A in 5 seconds?
Answer
(1) A→B→C→A
A→B→D→A
A→C→B→A
A→C→D→A
A→D→B→A
A→D→C→A
B : 7 ways, C : 7 ways , D : 7 ways
(2) 21 ways
(3) 60 ways
A→B→D→A
A→C→B→A
A→C→D→A
A→D→B→A
A→D→C→A
B : 7 ways, C : 7 ways , D : 7 ways
(2) 21 ways
(3) 60 ways
