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.

No. Eliminate the even numbers and those ending in 5. The sum of the digits in each of the remaining numbers will always produce a multiple of 3, so the numbers themselves will always be divisible by 3.

The resourceful 1. Qe5 threatens 2. Qb8#, and now:

1. … Rxe5 2. b7#
1. … Rd6 2. Qe8#
1. … Re8 2. Qxe8#
1. … Rxb6+ 2. Nxb6#
1. … Nc6 2. b7#

Imagine taking a practice run in a car that has plenty of gas. Along the way the car will pick up exactly enough gas to take it once around the track, so it will complete the practice run with the same amount of fuel that it started with. If we keep an eye on the car’s fuel gauge, we’ll find that it reaches its lowest point as we arrive at one of the depots (call it x_i). Any additional gas in our tank at that point is unneeded; we know it will never be used. Now if we start at x_i with an empty tank we know we’ll have enough gas to get from each depot to the next, running out of gas just as we arrive back at x_i.