Mathfest 2018 Puzzles: Order Eight Cyclic Groupdoku

Here is one of the puzzles I handed out after my group theory talk at Mathfest 2018.  If you're not sure what a group is, don't worry.  This puzzle is over the cyclic group of order eight, and cyclic groups have an easy to explain operation.  If you can tell time then you'll be good. 

Think of normal addition between positive whole numbers, except when you are done you subtract as many multiples of eight as it takes to get a non-negative number less than eight. This is very much like adding time on a clock, where if it is 10 o'clock and you add 7 hours you get to 5 o'clock.  The only difference is that we wrap around using the number eight instead of the number twelve.

Each clue here represents the sum of the entries in the cage after this operation has been applied.  For example, if there is a clue of four in a cage with two cells, their entries could be zero and four, one and three, or five and seven.  It is true that both two and two, and six and six also add to four, but as this is a Latin square puzzle, we cannot have adjacent entries equal to the same number.  

You'll probably notice as you proceed that the main challenge here is in the L-shaped cages, which not only contain three numbers instead of two, but allow us to have two of the same number so long as it is not in the corner.  

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