1. However hot it is, he never turns on air conditioner to save electricity.

2. However hot it is, he never turns on the air conditioner to save electricity.

3. However hot it is, he never turns on an air conditioner to save electricity.

4. However hot it may be, he never turns on air conditioner to save electricity.

5. However hot may be, he never turns on the air conditioner to save electricity.

6. However hot it may be, he never turns on an air conditioner to save electricity.

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Which ones are grammatical? Which one are commonly used? Thank you for your help.

#2 is the most common

Even it is not punctuated correctly. There should be a comma after "conditioner." Without the comma, it appears that one would "turn on the air conditioner to save electricity."

#3 or #6 could also be used.

The others are incorrect.

I agree with Steve except for the comma. In my opinion, the only comma needed in each sentence is the one that's already included.

http://grammar.ccc.commnet.edu/grammar/commas.htm
Uses #1 through #7 are the most common comma uses. Be sure to check out #11.

Out of the six sentences provided, the grammatically correct and commonly used ones are:

1. However hot it is, he never turns on the air conditioner to save electricity.
2. However hot it may be, he never turns on the air conditioner to save electricity.

Both of these sentences are grammatically sound and convey the intended meaning effectively. The phrase "however hot it is" or "however hot it may be" sets up a condition (in this case, the temperature being hot) under which the person mentioned does not turn on the air conditioner in order to save electricity. The use of "the" before "air conditioner" is more common since it refers to a specific appliance that exists.

It's important to note that while sentences 3, 4, 5, and 6 may be technically grammatical, they are less commonly used and might sound awkward or less natural to native speakers.