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I was taught that for aldehydes and ketone to exhibit keto-enol tautomerism, it is essential that there must be at least one α-hydrogen.

For example, acetophenone and butan-2-one show keto-enol tautomerism, but benzaldehyde and benzophenone don't show keto-enol tautomerism, since they don't have any α-hydrogens.

Examples of carbonyl compounds that do or do not exhibit keto-enol tautomerism

However, I was subsequently told that the cyclic enone 6,6-dimethylcyclohex-2-en-1-one also exhibits tautomerism:

6,6-Dimethylcyclohex-2-en-1-one

How is this possible? It does not have any α-hydrogens (the vinylic proton cannot be enolised as it is orthogonal to the C=O π* orbital).

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If you have an α,β-double bond next to your carbonyl, a γ hydrogen can also take part in tautomerism. Take crotonaldehyde $\ce{H3C-CH=CH-CHO}$:

$$\ce{O=CH-CH=CH-CH3 <<=> HO-CH=CH-CH=CH2}$$

In general, a double bond is able to ‘extend’ the mesomeric properties of a carbonyl compound across two additional atoms. This is known as vinylogous; crotonaldehyde can form a vinylogous enol.

In the specific case of 6,6-dimethylcyclohex-2-en-1-one, these are the two tautomers:

Tautomeric forms of 6,6-dimethylcyclohex-2-en-1-one

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There should be possibility of an intramolecular acid-base reaction. Acid-base pair are usually Lewis acid-base but there may be others. Such molecules show tautomerism. There are a large number of tautomerisms that can be explained by this way.

Such as keto-enol (your problem too), oxime-nitroso tautomerism and many more.

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