Game Play question

Question

There is a game that has categories for groups of three, five, and seven players. What is the minimum number of rounds that must be played for ten players in each group to play ten rounds?

Answer 1

Hi El K, this is a fun problem! Let's try to keep things simple by looking at the categories separately at first.

In groups of 3: for 10 players to get exactly 10 turns each, there will have to be 100 turns in total. Each round, 3 players get 1 turn (3 turns altogether). So, let's divide the 100 turns by 3 to see how many rounds of the game it will take for everyone to get 10 turns. 100 divided by 3 equals 33 with a remainder of 1. That means 1 player misses out on a 10th turn if only 33 rounds are played. It will take a minimum of 34 rounds for all 10 players to get all 10 turns. 2 lucky players get 11 turns.

Here's the long way: let's name each player a letter from A to J: A B C D E F G H I J.

Now let's play the game:

I copied and pasted the 10 letters 10 times, and then pressed return after every 3 letters, then numbered the rows. This proves that you need 34 rounds.

1. ABC
2. DEF
3. GHI
4. JAB
5. CDE
6. FGH
7. IJA
8. BCD
9. EFG
10. HIJ
11. ABC
12. DEF
13. GHI
14. JAB
15. CDE
16. FGH
17. IJA
18. BCD
19. EFG
20. HIJ
21. ABC
22. DEF
23. GHI
24. JAB
25. CDE
26. FGH
27. IJA
28. BCD
29. EFG
30. HIJ
31. ABC
32. DEF
33. GHI
34. JAB

The same methods can be used for the other categories.

For groups of 5: 20 rounds are needed because 100 divided by 5 equals 20 with no remainder.

1. ABCDE
2. FGHIJ
3. ABCDE
4. FGHIJ
5. ABCDE
6. FGHIJ
7. ABCDE
8. FGHIJ
9. ABCDE
10. FGHIJ
11. ABCDE
12. FGHIJ
13. ABCDE
14. FGHIJ
15. ABCDE
16. FGHIJ
17. ABCDE
18. FGHIJ
19. ABCDE
20. FGHIJ

For groups of 7: 15 rounds are needed because 100 divided by 7 equals 14 with a remainder of 2.

1. ABCDEFG
2. HIJABCD
3. EFGHIJA
4. BCDEFGH
5. IJABCDE
6. FGHIJAB
7. CDEFGHI
8. JABCDEF
9. GHIJABC
10. DEFGHIJ
11. ABCDEFG
12. HIJABCD
13. EFGHIJA
14. BCDEFGH
15. IJABCDE

Now, we just need to add up the rounds for the 3 categories. 34 + 20 + 15 = 69 rounds total.

I hope you find this explanation helpful. :^)