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In one 30 minute session we came up with the following answers. We look forward to trying to answer the other questions in the future.

What can you say if the numbers are all odd? It must be shifted up from an even number by an odd number or a odd number by a odd number.

What about if they are all even? It must be an even number shifted up by an even number.

What can you say if the units digits are all identical? It must be a multiple of ten.

What if there are only two different units digits? It must be a multiple of five.

What can you say if the difference between two numbers is prime? The shifted consecutive terms alternate between odd and even