We want to build a road between two cities, A and B, that are separated by a river. We can build a bridge, but it must be perpendicular to the river’s banks, as shown. Where along the river’s length should we place the bridge if we want to minimize the total length of the road?

# Black and White

From T.R. Dawson, a logic problem posing as a chess puzzle. If pinned men do not check, how can White mate in two moves?

# Alternate Routes

How many pairs of prime numbers are there whose sum is 999?

# The Donjon Keep Window

In Henry Dudeney’s Canterbury Puzzles, Sir Hugh De Fortibus takes his chief builder to the walls of his donjon keep and points to a window there.

“Methinks,” he says, “yon window is square, and measures, on the inside, one foot every way, and is divided by the narrow bars into four lights, measuring half a foot on every side.”

“Of a truth that is so, Sir Hugh,” says the builder.

“Then I desire that another window be made higher up whose four sides shall also be each one foot, but it shall be divided by bars into eight lights, whose sides shall be all equal.”

Bewildered, the builder says, “Truly, Sir Hugh, I know not how it may be done.”

“By my halidame!” exclaims De Fortibus in pretended rage. “Let it be done forthwith. I trow thou art but a sorry craftsman, if thou canst not, forsooth, set such a window in a keep wall.”

How can it be done?

# Black and White

Billiards chess is a variant of traditional chess in which the pieces carom off the sides of the board at right angles. In the diagram above, the white bishop at a2 controls the diagonal a2-g8 as in normal chess, but its attack also “bounces” from g8 to h7 and then back along the h7-b1 diagonal. Both bishops attack and move along these “bouncing” lines. How can White mate the black king in two moves?

# Speaking Volumes

A puzzle by A. Kozlov from the Soviet popular science magazine Kvant:

Watching a solar eclipse, a girl asks her father how much farther away is the sun than the moon.

He says, “As far as I remember, 387 times farther.”

She says, “Then I can figure out how much greater is the sun’s volume than the moon’s.”

# Royal Descent

A puzzle from Stuart Collingwood’s Lewis Carroll Picture Book, 1899:

A captive Queen and her son and daughter were shut up in the top room of a very high tower. Outside their window was a pulley with a rope round it, and a basket fastened at each end of the rope of equal weight. They managed to escape with the help of this and a weight they found in the room, quite safely. It would have been dangerous for any of them to come down if they weighed more than 15 lbs. more than the contents of the lower basket, for they would do so too quick, and they also managed not to weigh less either.

The one basket coming down would naturally of course draw the other up.

The Queen weighed 195 lbs., daughter 105, son 90, and the weight 75.

How did they do it?

# Black and White

By Oskar Blumenthal. White to mate in two moves.

# Black and White

One more chess curiosity by T.R. Dawson: How can White mate in two half moves?

The answer is to play the first half of Bg1-f2, and the second half of Bf1-g2, thus getting the white bishop from g1 to g2 and giving mate.

A fair-minded reader might ask why Black can’t pull the same trick, transferring his bishop from b8 to b7 to block the check. The answer, Dawson argues, is that some of the constituent moves are illegal: Black can’t combine Bb8-c7 and Bc8-b7 because a bishop on c8 would put the white king in an unreal check on h3; and he can’t combine Bb8-a7 and Ba8-b7 because a8 is occupied.

From Caissa’s Fairy Tales (1947).

# Crossing Guard

Suppose some 2n points in the plane are chosen so that no three are collinear, and then half are colored red and half blue. Will it always be possible to connect each red point with a blue one, in pairs, so that none of the connecting lines intersect?