1. Fogg and Passepartout go to Allahabad by elephant.

2. Fogg and Passepartout go to Allahabad by the elephant.

3. Fogg and Passepartout go to Allahabad on an elephant.

4. Fogg and Passepartout go to Allahabad on the elephant.
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Which one is grammatical?

All are grammatical except for 1.

Using "the" suggests the elephant is a specific elephant already identified. "An" elephant is a single elephant, but not one in particular. "On" and "by" mean much the same thing. 1 and 3 are best, unless the elephant has already been identified by name or some other specific way.

Ms. Sue and I disagree on #1. ????

I made a mistake. 1 is grammatical. Thanks for catching this, Reed.

All four options:

1. Fogg and Passepartout go to Allahabad by elephant.
2. Fogg and Passepartout go to Allahabad by the elephant.
3. Fogg and Passepartout go to Allahabad on an elephant.
4. Fogg and Passepartout go to Allahabad on the elephant.

are grammatically correct. However, the slight difference between them lies in the level of specificity or generality.

Option 1 implies that Fogg and Passepartout are traveling using any elephant available, without specifying a particular elephant. It suggests a more general statement about their mode of transportation.

Option 2, on the other hand, specifies that they are using a specific elephant, likely one that has been previously mentioned or is known to the speaker and listener.

Option 3 uses the preposition "on" and the indefinite article "an" to indicate that they are using any elephant for their journey to Allahabad.

Option 4 specifies that they are using a specific elephant for their journey to Allahabad.

So, the choice between these options will depend on the context and level of detail you want to convey in your statement.