How can I SELECT rows with MAX(Column value), PARTITION by another column in MYSQL?

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You are so close! All you need to do is select BOTH the home and its max date time, then join back to the topten table on BOTH fields:

SELECT tt.*
FROM topten tt
INNER JOIN
    (SELECT home, MAX(datetime) AS MaxDateTime
    FROM topten
    GROUP BY home) groupedtt 
ON tt.home = groupedtt.home 
AND tt.datetime = groupedtt.MaxDateTime

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