Difference between revisions of "2004 AMC 12A Problems/Problem 7"
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+ | {{duplicate|[[2004 AMC 12A Problems|2004 AMC 12A #7]] and [[2004 AMC 10A Problems/Problem 8|2004 AMC 10A #8]]}} | ||
==Problem== | ==Problem== | ||
− | A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token to each of the other players and also places one token in the discard pile. The game ends when some player runs out of tokens. Players <math>A</math>, <math>B</math>, and <math>C</math> start with 15, 14, and 13 tokens, respectively. How many rounds will there be in the game? | + | A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token to each of the other players and also places one token in the discard pile. The game ends when some player runs out of tokens. Players <math>A</math>, <math>B</math>, and <math>C</math> start with <math>15</math>, <math>14</math>, and <math>13</math> tokens, respectively. How many rounds will there be in the game? |
<math> \mathrm{(A) \ } 36 \qquad \mathrm{(B) \ } 37 \qquad \mathrm{(C) \ } 38 \qquad \mathrm{(D) \ } 39 \qquad \mathrm{(E) \ } 40 </math> | <math> \mathrm{(A) \ } 36 \qquad \mathrm{(B) \ } 37 \qquad \mathrm{(C) \ } 38 \qquad \mathrm{(D) \ } 39 \qquad \mathrm{(E) \ } 40 </math> | ||
− | == | + | ==Solutions== |
− | |||
− | Therefore, after 12 sets | + | ===Solution 1=== |
+ | We look at a set of three rounds, where the players begin with <math>x+1</math>, <math>x</math>, and <math>x-1</math> tokens. | ||
+ | After three rounds, there will be a net loss of <math>1</math> token per player (they receive two tokens and lose three). Therefore, after <math>36</math> rounds -- or <math>12</math> three-round sets, <math>A,B</math> and <math>C</math> will have <math>3</math>, <math>2</math>, and <math>1</math> tokens, respectively. After <math>1</math> more round, player <math>A</math> will give away <math>3</math> tokens, leaving them empty-handed, and thus the game will end. We then have there are <math>36+1=\boxed{\mathrm{(B)}\ 37}</math> rounds until the game ends. | ||
+ | |||
+ | ===Solution 2=== | ||
+ | |||
+ | Let's bash a few rounds. The amounts are for players <math>1,2,</math> and <math>3</math>, respectively. | ||
+ | |||
+ | First round: <math>15,14,13</math> (given) | ||
+ | Second round: <math>12,15,14</math> | ||
+ | Third round: <math>13,12,15</math> | ||
+ | Fourth round: <math>14,13,12</math> | ||
+ | |||
+ | We see that after <math>3</math> rounds are played, we have the exact same scenario as the first round but with one token less per player. So, the sequence <math>1,4,7,10...</math> where each of the next members are <math>3</math> greater than the previous one corresponds with the sequence <math>15,14,13,12...</math> where the first sequence represents the round and the second sequence represents the number of tokens player <math>1</math> has. But we note that once player <math>1</math> reaches <math>3</math> coins, the game will end on his next turn as he must give away all his coins. Therefore, we want the <math>15-3+1=13</math>th number in the sequence <math>1,4,7,10...</math> which is <math>\boxed{\mathrm{(B)}\ 37}</math>. | ||
+ | |||
+ | Solution by franzliszt | ||
== See also == | == See also == | ||
+ | {{AMC12 box|year=2004|ab=A|num-b=6|num-a=8}} | ||
{{AMC10 box|year=2004|ab=A|num-b=7|num-a=9}} | {{AMC10 box|year=2004|ab=A|num-b=7|num-a=9}} | ||
+ | {{MAA Notice}} |
Revision as of 20:31, 16 July 2020
- The following problem is from both the 2004 AMC 12A #7 and 2004 AMC 10A #8, so both problems redirect to this page.
Problem
A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token to each of the other players and also places one token in the discard pile. The game ends when some player runs out of tokens. Players , , and start with , , and tokens, respectively. How many rounds will there be in the game?
Solutions
Solution 1
We look at a set of three rounds, where the players begin with , , and tokens. After three rounds, there will be a net loss of token per player (they receive two tokens and lose three). Therefore, after rounds -- or three-round sets, and will have , , and tokens, respectively. After more round, player will give away tokens, leaving them empty-handed, and thus the game will end. We then have there are rounds until the game ends.
Solution 2
Let's bash a few rounds. The amounts are for players and , respectively.
First round: (given) Second round: Third round: Fourth round:
We see that after rounds are played, we have the exact same scenario as the first round but with one token less per player. So, the sequence where each of the next members are greater than the previous one corresponds with the sequence where the first sequence represents the round and the second sequence represents the number of tokens player has. But we note that once player reaches coins, the game will end on his next turn as he must give away all his coins. Therefore, we want the th number in the sequence which is .
Solution by franzliszt
See also
2004 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 6 |
Followed by Problem 8 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
2004 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.