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A billiard rack is a rack, usually a triangle, that can hold a certain number of equal size billiard balls, such that the balls' centres cannot move within the rack.

Can the rack be a square, for every number of balls? (For 1 to 25 balls, the answer is yes.)

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    $\begingroup$ There is something I don't understand. I'm really naive, but if you do it for n balls, can't you also do it for 4n balls? I mean why the 8 ball rack is not 4 times the 4 ball rack?? $\endgroup$
    – Marsault Chabat
    Commented Jan 6, 2023 at 13:35
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    $\begingroup$ @MarsaultChabat: I think the pictures in the link try to maximize the size of the square, hence the "new" construction for $n=8$. $\endgroup$
    – Joachim König
    Commented Jan 6, 2023 at 14:05
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    $\begingroup$ @MarsaultChabat There exists a square rack that can hold 8 balls, so there exists a square rack that can hold 32 balls, but I'm not sure if there's a square rack that can hold 31 balls. If you just remove one ball from the rack with 32 balls, then the other balls are free to move. $\endgroup$
    – Dan
    Commented Jan 6, 2023 at 14:41
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    $\begingroup$ It's worth clarifying that "the balls' centres cannot move within the rack" means that motion is impossible even if the balls move simultaneously. This is different from saying that no individual ball can be moved without disturbing other balls. $\endgroup$
    – Timothy Chow
    Commented Jan 7, 2023 at 0:33
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    $\begingroup$ @JoachimKönig Gluing mirror images does not always result in a rigid structure. For example, if you take the 2 ball rack, there are two ways to glue mirror images: one produces a sort of X formation, the other produces a sort of ring. In the X-formation, the balls can move (just rotate the 4 balls in the centre about their collective centre). In the ring formation, the balls can move (see here for an explanation of a similar fact). $\endgroup$
    – Dan
    Commented Jan 7, 2023 at 1:38

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