Like the 1823 Patched 3, the Ugly 3 is a modification (or late state) of a die that resulted in a mal-formed 3. In the case of the Ugly 3, the 3 started out acceptably, but some unknown mint engraver attempted to thicken the juncture between the top and lower curves of the 3, but the attempt was so crude and out-of-hand that the 3 became simply ugly.
This variety is more scarce than the Patched 3 and is actually rather difficult to locate in Mint State. One of the nicest examples and certainly one of the most attractive is the PCGS MS65 from the D. Brent Pogue Collection.
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