The following puzzle has just one solution.
TOUGH
+ TOUGH
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PUZZLE
Can you find it?
... and this one has two solutions.
TOUGH
TOUGH
+ TOUGH
------
PUZZLE
Can you find them?
First, it's obvious that P=1, and T is at least 5.
Since the O+O and U+U columns have a Z at the bottom, either G+G is greater than 10 and U+U is also greater than 10, or G+G is less than 10 and U+U is also less than 10.
1st case: G greater than 5. U=5, O=0, Z=1; doesn't work. U=6, O=1; doesn't work. U=7, O=2, Z=5; T=8; doesn't work. U=8, O=3, Z=7, T=9... G is 5 or 6, and H is 2, 4, 5 or 6. A little trial and error shows G=5 and H=2 works: 93852+93852=187704.
2nd case: why bother? If there's only one solution, this must not work out.
First solution
