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Omni-MATH / You start out with a big pile of 3²⁰⁰⁴ cards, with the numbers 1,2,3,…,3²⁰⁰⁴ written on them. You arrange the cards into groups of three any way you like; from each group, you keep…

Problem

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problem

You start out with a big pile of 320043^{2004} cards, with the numbers 1,2,3,…,320041,2,3, \ldots, 3^{2004} written on them. You arrange the cards into groups of three any way you like; from each group, you keep the card with the largest number and discard the other two. You now again arrange these 320033^{2003} remaining cards into groups of three any way you like, and in each group, keep the card with the smallest number and discard the other two. You now have 320023^{2002} cards, and you again arrange these into groups of three and keep the largest number in each group. You proceed in this manner, alternating between keeping the largest number and keeping the smallest number in each group, until you have just one card left. How many different values are possible for the number on this final card?
Plain-text mathematical notation (without MathML)
You start out with a big pile of 3²⁰⁰⁴ cards, with the numbers 1,2,3,…,3²⁰⁰⁴ written on them. You arrange the cards into groups of three any way you like; from each group, you keep the card with the largest number and discard the other two. You now again arrange these 3²⁰⁰³ remaining cards into groups of three any way you like, and in each group, keep the card with the smallest number and discard the other two. You now have 3²⁰⁰² cards, and you again arrange these into groups of three and keep the largest number in each group. You proceed in this manner, alternating between keeping the largest number and keeping the smallest number in each group, until you have just one card left. How many different values are possible for the number on this final card?
Original LaTeX notation
You start out with a big pile of $3^{2004}$ cards, with the numbers $1,2,3, \ldots, 3^{2004}$ written on them. You arrange the cards into groups of three any way you like; from each group, you keep the card with the largest number and discard the other two. You now again arrange these $3^{2003}$ remaining cards into groups of three any way you like, and in each group, keep the card with the smallest number and discard the other two. You now have $3^{2002}$ cards, and you again arrange these into groups of three and keep the largest number in each group. You proceed in this manner, alternating between keeping the largest number and keeping the smallest number in each group, until you have just one card left. How many different values are possible for the number on this final card?

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Source and history

task_id
897614fe-af98-5e6d-a2c7-e793288724ef
task_key
test--897614fe-af98-5e6d-a2c7-e793288724ef
task_revision_id
1
config / split
/ test

Official source

initial import