The Hatter is telling the truth. This is the only possibility that doesn’t lead to a contradiction. The Dodo lies when he says that the Hatter lies; the Hatter says correctly that the March Hare lies; and the March Hare lies when he says that both the Dodo and the Hatter lie (in fact, only the Dodo does).

1. Qa8! and the queen or the rook must mate.

From E.B. Cook et al., American Chess-Nuts, 1868.

Vowels.

B is taller:

• If A and B stand in the same row, then B is taller, since we know that B is the tallest student in his row.
• If A and B stand in the same column, then again B is taller, since we know that A is the shortest student in his column.
• If A and B share neither a row nor a column, then let C be the student who’s in the same column as A and the same row as B. Then C is shorter than B (who is the tallest in his row) and taller than A (who is the shortest in his column), so B > C > A.

In every case, then, B is taller than A.

1. Nb7, followed by 2. Qe4#.

From Le Palamède, via Charles Tomlinson, Amusements in Chess, 1845.

If n is odd then the sequence (1, 2, 3, … n) begins and ends with an odd number. This means it contains one more odd number than even. Likewise (a₁, a₂, … a_n), which are only the same numbers in a different order. This means that two odd numbers must end up in the same bracket — there are not enough even numbers available to pair each with an odd. And if two odd numbers appear in the same bracket, they’ll produce an even difference and hence an even product.

From Ross Honsberger, Ingenuity in Mathematics, 1970.

Three. Three men can be mutual brothers-in-law if A marries B’s sister, B marries C’s sister, and C marries A’s sister. But D has no way to join this group. He cannot ask his sister to marry A, B, or C, because they are already spoken for. And if he himself marries the sister of, say, A, then he cannot be a brother-in-law to C unless C or D is A’s brother, which implies a violation of the law against consanguinity. (If a man may marry his own sister, then there’s no limit to the number of mutual brothers-in-law.)

(David L. Silverman, “A Problem of Relations,” Journal of Recreational Mathematics 3:3, July 1970)

They’ll form a circle:

1. Qd2 and Black can’t avoid a fatal discovered check.

Two widow Ladies married were,
Each to the other’s Son;
And they both pregnant did appear
E’er one full Year was run.
The Consequence of which did prove,
To each a charming Boy;
This did cement their Husband’s Love,
And added to their Joy.
By this Event likewise it’s plain
They did commence Grandmothers,
And that their Husbands did obtain
Two young delightful Brothers.
A Brother you may justly call
Each, to the other’s Father:
Uncles they were reciprocal,
You easily may gather.