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In Gin, What Happens When One Calls But the Opposing Player has A Lower Call Card?

Eg. The call card is 10, and I call placing everything but a 6. My opponent didn't call, but was able to place everything but a 4.

So, my opponent had a lesser amount of points she couldn't place than I, but I called.

What does one do regarding points when this happens? Does she get the difference, or does no-one get points? Or does she get points like she would if she had successfully gotten a Gin?

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Your opponent Undercut you and scores 10 points plus the difference between your score and theirs (+2 in your case). Actually, unless you knockout on gin (zero points), if your opponent ties your score you are undercut. From pagat.com:

If the knocker did not go gin, and the counts are equal, or the knocker's count is greater than that of the opponent, the knocker has been undercut. In this case the knocker's opponent scores the difference between the counts plus a 10 point bonus.

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From Wikipedia: Undercut (or underknocking) Occurs when the defending player has a deadwood count lower than or equal to that of the knocking player (this can occur either naturally or by laying off after a knock). In this case, the defender scores an undercut bonus of 25 points plus the difference between the two hands. (In some rule sets, the bonus is only 10 or 20 points, or is not awarded in case of a tie.) –  Chris Persichetti Apr 25 '13 at 15:38
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