Geometry: An Interactive Journey to Mastery

3. Color the corners of the squares of the table
   black and white according to a checkerboard
   pattern as shown in Figure S.27.1, with the
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   It is clear that the ball will only ever pass from
   black corner to black corner. So, if the ball is to
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   corner of the table, the only black corner.
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   To prove that the ball must land in a corner, we need to establish two things.
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   7KHPRWLRQRIWKHEDOOLVWLPHUHYHUVLEOH,IZHVHHLWWUDYHUVHRQHVTXDUHRIWKHWDEOHZHNQRZSUHFLVHO\
   from which square it just came. So, if the ball traverses some square more than one time, the previous
   square in the ball’s path was also repeated more than once, and so on.
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   the ball will never pass through the same square of the table twice. This means that the ball cannot bounce
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   UHTXLUHVWUDYHUVLQJWKHWRSOHIWVTXDUHPRUHWKDQRQFH௘ͽ7KHEDOOWKHUHIRUHPXVWIDOOLQWRRQHRIWKH
   remaining three corners.
   Comment: This conclusion that the ball falls into one of the remaining three corners applies to tables
   of all dimensions.


4. :LWKWKHFRORULQJVFKHPHLPSOLHGE\3UREOHPRQO\WKHERWWRPOHIWFRUQHULVEODFNLQHYHQîRGGWDEOHV
   This is the corner into which the ball must fall.


5. :LWKWKHFRORULQJVFKHPHLPSOLHGE\3UREOHPRQO\WKHWRSULJKWFRUQHULVEODFNLQRGGîHYHQWDEOHV
   This is the corner into which the ball must fall.

Figure S.27.1