INMO - 2003

Probability Level pending

In a lottery, tickets are given nine-digit numbers using only the digits $1,2,3$ . They are also colored red, blue or green in such a way that two tickets whose numbers differ in all the nine places get different colors.

Suppose the ticket bearing the number $222222222$ is green & that bearing the number $122222222$ is red, then determine the color of the ticket bearing the number $123123123$ .

Green Help! I can't solve it! Blue Red

1 solution

Shishir Shahi
Jun 25, 2017

Observe the first number (left-most) number.

Let the color of ticket 123123123 be x. The color of 331331331 must be y, where x and y are blue and green in some order, because the position of digits in 331331331 is different than that of 122222222 and 123123123. Also observe that 123123123 could be red, because it has the same 1st digit as 122222222. Now observe, 331331331 has different digits than 222222222 at all places, and therefore the color of 331331331 cannot be green. Also, it cannot be red, since it has different digits than 122222222 at all places. Therefore, the color of 331331331 is blue.Therefore, the color of 123123123 is either red or green. Now consider 231331331 and 313113113. The latter must have blue color, and therefore the previous one must have green color and since 123123123 and 231331331 cannot have the same color, the color of 123123123 has to be Red. I hope this messy solution gets upvoted.

Alex Fullbuster - 2 years, 1 month ago

Shishir Shahi Wrong! This is an invalid thought, you need to justify it.

Alex Fullbuster - 2 years, 1 month ago

INMO - 2003

1 solution

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