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Omni-MATH / Suppose that k≥2 is a positive integer. An in-shuffle is performed on a list with 2k items to produce a new list of 2k items in the following way: - The first k items from the orig…

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problem

Suppose that k≥2k \geq 2 is a positive integer. An in-shuffle is performed on a list with 2k2 k items to produce a new list of 2k2 k items in the following way: - The first kk items from the original are placed in the odd positions of the new list in the same order as they appeared in the original list. - The remaining kk items from the original are placed in the even positions of the new list, in the same order as they appeared in the original list. For example, an in-shuffle performed on the list PQRSTUP Q R S T U gives the new list PSQTRUP S Q T R U. A second in-shuffle now gives the list PTSRQUP T S R Q U. Ping has a list of the 66 integers from 1 to 66, arranged in increasing order. He performs 1000 in-shuffles on this list, recording the new list each time. In how many of these 1001 lists is the number 47 in the 24th position?
Plain-text mathematical notation (without MathML)
Suppose that k≥2 is a positive integer. An in-shuffle is performed on a list with 2k items to produce a new list of 2k items in the following way: - The first k items from the original are placed in the odd positions of the new list in the same order as they appeared in the original list. - The remaining k items from the original are placed in the even positions of the new list, in the same order as they appeared in the original list. For example, an in-shuffle performed on the list PQRSTU gives the new list PSQTRU. A second in-shuffle now gives the list PTSRQU. Ping has a list of the 66 integers from 1 to 66, arranged in increasing order. He performs 1000 in-shuffles on this list, recording the new list each time. In how many of these 1001 lists is the number 47 in the 24th position?
Original LaTeX notation
Suppose that $k \geq 2$ is a positive integer. An in-shuffle is performed on a list with $2 k$ items to produce a new list of $2 k$ items in the following way: - The first $k$ items from the original are placed in the odd positions of the new list in the same order as they appeared in the original list. - The remaining $k$ items from the original are placed in the even positions of the new list, in the same order as they appeared in the original list. For example, an in-shuffle performed on the list $P Q R S T U$ gives the new list $P S Q T R U$. A second in-shuffle now gives the list $P T S R Q U$. Ping has a list of the 66 integers from 1 to 66, arranged in increasing order. He performs 1000 in-shuffles on this list, recording the new list each time. In how many of these 1001 lists is the number 47 in the 24th position?

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Source and history

task_id
6d5299c9-c1a0-5056-a81c-99d91626b307
task_key
test--6d5299c9-c1a0-5056-a81c-99d91626b307
task_revision_id
1
config / split
/ test

Official source

initial import