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Rotating round table

Source: Mathematical Circus by Martin Gardner

At a large dinner, 24 people are to sit evenly spaced around a round table. Place cards are laid to show where everyone should sit. Unfortunately nobody notices the name cards and the guests sit down with nobody in the correct seat.

Show that it is possible to rotate the table so that at least two people will be in the correct seats.

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Let

p (n)

denote the number of people in the correct seat when the table is rotated by

n

people. From the question, we know that:

p (0) = 0

As every guest can be put into the correct place by one rotation, we know that:

p (0) + p (1) + . . . + p (23) = 24

As

p (0) = 0

:

p (1) + . . . + p (23) = 24

This sum has 23 positive integers adding up to 24, so one of the rotations must lead to at least two people being in the correct places.

Extension

If the 24 guests sat randomly and one person was in the correct seat, could you still rotate the table so that two people are correctly seated?

If you enjoyed this puzzle, check out Sunday Afternoon Maths XLIV,
puzzles about numbers, or a random puzzle.

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