Question

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 ____.

Answer 1

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
<b>since the car toys are in position 1 and 2, they are held by C and E</b>

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 )