Sort of Musical Chairs Math Task day 5

12 people in a line of chairs

When the round starts everyone either moves one chair to the left or right or they stay put. So the players rearrange – 1 per chair but they can move at most one chair. HOW MANY WAYS CAN THIS BE DONE?

HERE IS AN EXAMPLE

How Many with 5? ABCDE→BACDE ABCDE→BADCE ABCDE→BACED ABCDE→ABCDE ABCDE→ACBDE ABCDE→ACBED ABCDE→ABDCE ABCDE→ABCED Can you classify these? A MOVES A FIXED

Let A n be the number of ways to rearrange n-players What is A 1 ? What is A 2 ? What is A 3 ? What is A 4 ? We know that A 5 =8

A n =A n-1 +A n-2 FIRST GUY MOVES FIRST GUY FIXED A************************ BA********************** N-1 N-2

So what is A 12 ? A 6 = A 5 + A 4 =8+5=13 A 7 = A 6 + A 5 =13+8=21 A 8 = A 7 + A 6 =21+13=34 A 9 = A 8 + A 7 =34+21=55 A 10 = A 9 + A 8 =55+34=89 A 11 = A 10 + A 9 =89+55=144 A 12 = A 11 + A 10 =144+89=233

Same Rules but now IN A CIRCLE B 12 =?

(1)Everybody Moves Right (2)Everybody Moves Left (3)Ed stays Fixed (4)Ed Swaps Right (5)Ed Swaps Left (1)1 way (2)1 way (3)A 11 ways (4)A 10 ways (5)A 10 ways B 12 = = 324 ED