Problem 1

Three players

A

B

, and

C

play the following game. On each of three cards an integer is written; the three numbers

p, q, r

satisfy

0 < p < q < r

. The three cards are shuffled and one is dealt to each player, and each player receives as many counters as the number on the card they hold. The cards are shuffled and dealt again, and this is repeated for at least two rounds (counters from earlier rounds stay with the players). After the last round,

A

has

20

counters in all,

B

has

10

, and

C

has

9

. In the last round,

B

received

r

counters. Who received

q

counters on the first round?

Step 4 of 6: Rule out

r=7

Previous step Next step

r=7,\quad 10-r=3\implies p+q=3\implies p+q+r=10\ne13

Detailed analysis

If $r=7$ , the two cards B received before the last round sum to $10-7=3$ . Neither card can be $r$ , since then the sum would be at least $7+1>3$ . The two cards are therefore $p$ and/or $q$ . The possibility $2p=3$ is impossible for integer $p$ , while $2q\ge4$ ; hence they must be $p+q=3$ , giving $p=1,q=2$ and $p+q+r=10$ , contradicting Step 2. Thus $r\ne7$ .