Here are some examples:
* It forgets the rules of the Caucus-race: "The Dodo suddenly called out 'The race is over!' and they all crowded around it, panting. 'But who has won?' asked Alice, getting up and brushing herself. 'That's the great puzzle,' said the Dodo, 'because everyone has won, and all must have prizes.'
* It suggests nonsensical activities: "Everybody has won, and all must have prizes!' cried the Dodo. 'But who is to give the prizes?' asked Alice. 'Why, she, of course,' said the Dodo pointing to Alice with its bill. 'And she must be the judge, too!'
* It struggles to count: "Then the Dodo solemnly got up and said, 'I think, considering the amount of ground we have gone over, that I ought to have a little present.' 'Of course you ought,' said Alice. 'And you ought to have it now.' 'Certainly,' said the Dodo, 'but I must be the judge, you know.' So they all sat down again in a circle, and the Dodo said very gravely, 'Ahem! Now, I'll give the prizes. First prize goes to... let me see... (Here the Dodo began to think again.)
While the Dodo's overall intelligence is not a central theme in the story, its clumsiness and forgetfulness contribute to the humor and absurdity of the scene. It is more of a symbol of the nonsense and illogical nature of the world Alice encounters than a character with notable intelligence.