5 kids (A, B, C, D and E) are playing games in the playground. In each round, they follow the rules below to pass on the toy on their hands to the next kid: A→C, B→E, C→A, D→B, E→D. Initially, A and B are holding the car toy each while C, D and E holding motorcycle toy each. After five rounds, the kids with car toy are ____ and ____.
Think of the 5 kids standing in a row, and in each row changing positions
1 ---> 3
2 ---> 5
3 ---> 1
4 ---> 2
5 ---> 4
At the start,,,: 1 2 3 4 5 ---> A B C D E
after 1 round: 3 4 1 5 2 ---> C D A E B
after 2 round: 1 5 3 2 4 ---> A E C B D
after 3 round: 3 2 1 4 5 ---> C B A D E
after 4 round: 1 4 3 5 2 ---> A D C E B
after 5 round: 3 5 1 2 4 ---> C E A B D
since the car toys are in position 1 and 2, they are held by C and E
did you notice that column 1 and 3 are interchanged, with A in position 1 and B in 3 for each even round and B in position 1 and A in 3 for each odd round.
Column 2 rotates as BDEBDEBDE...
column 4 rotates as DEBDEBDEB...
column 5 rotates as EBDEBDEBD
The order of BDE in those three columns never changes, they just sequence with a different starting order.
Thus we could establish the position after every round, by considering divisibility by 2's and 3's
(the A..C are back to the original after 2 rounds, the B..D..E are back in the original order after 3 rounds.
So after every 6 rounds the original order is returned )
To determine which kids have the car toy after five rounds, we need to follow the rules for passing on the toy and keep track of the changes.
Initial setup:
A and B have the car toy,
C, D, and E have the motorcycle toy.
Round 1:
A passes the car toy to C.
B passes the car toy to E.
Result:
C has the car toy,
E has the car toy,
A, B, and D still have the motorcycle toy.
Round 2:
C passes the car toy to A.
E passes the car toy to D.
Result:
A and C have the car toy,
B and E have the motorcycle toy,
D still has the motorcycle toy.
Round 3:
A passes the car toy to C.
D passes the car toy to B.
Result:
C has the car toy,
B has the car toy,
A, D, and E still have the motorcycle toy.
Round 4:
C passes the car toy to A.
B passes the car toy to E.
Result:
A and E have the car toy,
B has the motorcycle toy,
C and D still have the motorcycle toy.
Round 5:
A passes the car toy to C.
E passes the car toy to D.
Result:
C and D both have the car toy,
B has the motorcycle toy,
A and E still have the motorcycle toy.
Therefore, after five rounds, the kids with the car toy are C and D.
To determine which kids have the car toy after five rounds, we need to go through each round and track the movement of the toys. Let's start with the initial setup:
A: Car B: Car C: Motorcycle D: Motorcycle E: Motorcycle
Round 1:
A passes the car toy to C, so the new arrangement is:
A: X B: Car C: Car D: Motorcycle E: Motorcycle
Round 2:
B passes the car toy to E, so the new arrangement is:
A: X B: X C: Car D: Motorcycle E: Car
Round 3:
C passes the car toy to A, so the new arrangement is:
A: Car B: X C: X D: Motorcycle E: Car
Round 4:
D passes the car toy to B, so the new arrangement is:
A: Car B: Car C: X D: X E: Car
Round 5:
E passes the car toy to D, so the final arrangement is:
A: Car B: Car C: X D: Car E: X
Therefore, after five rounds, the kids with the car toy are A and B.