1998 AHSME Problem 20

Number Theory Level 3

Three cards, each with a positive integer written on it, are lying face-down on a table. Casey, Stacy, and Tracy are told that

(a) the numbers are all different

(b) they sum to 13

First, Casey looks at the number on the leftmost card and says, "I don't have enough information to determine the other two numbers." Then Tracy looks at the number on the rightmost card and says, "I don't have enough information to determine the other two numbers." Finally, Stacy looks at the number on the middle card and says, "I don't have enough information to determine the other two numbers." Assume that each person knows that the other two reason perfectly and hears their comments. What number is on the middle card?

There is not enough information to determine the number 3 4 5

1 solution

Henry Wan
Oct 22, 2015

$It\quad is\quad not\quad hard\quad to\quad check\quad that\quad there\quad are\quad only\quad 8\quad possible\quad outcomes,\quad which\quad are\\ (1,2,10),\quad (1,3,9),\quad (1,4,8),\quad (1,5,7)\\ (2,3,8),\quad (2,4,7),\quad (2,5,6)\\ (3,4,6)\\ \\ Casey's\quad IDK\quad rules\quad out\quad (3,4,6)\quad \\ because\quad only\quad 1\quad possible\quad outcome\quad with\quad 3\quad on\quad the\quad leftmost\quad card.\\ \\ Tracy's\quad IDK\quad rules\quad out\quad (2,5,6),\quad (1,2,10)\quad and\quad (1,3,9)\\ because\quad only\quad 1\quad possible\quad outcome\quad with\quad that\quad number\quad on\quad the\quad rightmost\quad card.\\ \\ Stacy's\quad IDK\quad rules\quad out\quad (1,5,7)\quad and\quad (2,3,8).\\ \\ The\quad remaining\quad possible\quad outcomes\quad (1,4,8)\quad and\quad (2,4,7)\quad both\quad have\quad 4\quad on\quad the\quad middle\quad card.$

1998 AHSME Problem 20

1 solution

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