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Everyone has won and all must have prizes. Thus, the solution is to distribute the prizes so that each participant receives one prize.

Each character represents a different number. The solution is H=8, A=1, T=0, E=5, R=2. The total sum is 10852.

The final count of shapes is 20 circles, 15 squares, and 10 triangles.

The pocket watch indicates time at a speed where one Wonderland hour equals 12 real-world hours. Thus, in 2 Wonderland hours, 24 real-world hours pass.

The tale is a tale of 'tails', a sequence of numbers that follow a pattern. The pattern is that each number in the sequence represents the length of the previous 'tail', starting with 3.

Drinking from the bottle makes Alice shrink to a tenth of her size, and eating the cake makes her grow 10 times her current size.

Only the Cheshire Cat's grin remains because if all cats are grinning and a grin always means happiness, then all cat's grins signify happiness, hence the Cheshire Cat's grin remains.

The sequence for striking the hedgehogs is based on the Fibonacci sequence (1, 1, 2, 3, 5...). The next number is the sum of the previous two numbers of hedgehogs struck.

The paradox is that if they agree never to disagree, then they cannot have an argument, which means they are agreeing, which violates their agreement not to agree.

The answer to the arithmetic problem in the poem is 4. The operations in the riddle are non-standard and follow a special logic within the poem's context.

The solution is that the riddle 'Why is a raven like a writing-desk?' has no answer. It is simply meant to be pondered upon.