We have a strong intuition that it’s wrong to punish someone for a crime he hasn’t committed, but is this always the case? Algy, an Alaskan motorist, and Ben, a traffic policeman, both know reliably that Algy intends to speed on a remote, unpatrolled, but radar-surveyed highway at 10:31 tomorrow morning. They also know that, if this happens, police won’t be able to reach the scene of the offense until several hours after it’s been committed.

Algy radios Ben with an offer: If Ben issues a ticket for this crime before it occurs, Algy will pay the fine. If Ben doesn’t issue the ticket until after the offense has occurred, then Algy will flee the country to avoid paying the fine. Ben thinks about this, then issues the ticket now, writing in tomorrow’s date. He delivers the ticket to Algy’s address, where Algy’s wife gives him Algy’s check for the fine, which Ben cashes immediately. At 10:31 tomorrow, Algy exceeds the speed limit as described.

University of Hong Kong philosopher Christopher New writes, “If this example is valid, it suggests that there may be room in our moral thought for the notion of prepunishment [punishment before an offense is committed], and that it may be only epistemic, rather than moral, constraints that prevent us from practising it.”

(Christopher New, “Time and Punishment,” Analysis 52:1 [January 1992], 35-40.)

An ambiguous 1904 lithograph, via the Library of Congress.

Somewhat similar: the spinning dancer.

From the Strand, May 1899:

Our next photograph is a facsimile of an address on a letter that found its way from Spain to the G.P.O., St. Martin’s-le-Grand. Remarkable as it may seem, this specimen of handwriting was deciphered by ‘the blind man of St. Martin’s,’ and the letter safely reached its destination. It is addressed to the ‘Spanish Ambassador (or Embassy), London.’

(The “blind men” were “the decipherers of illegible and imperfect addresses” in the returned letter office of the General Post Office. Further examples.)

In Bologna, the former convent of San Michele in Bosco contains a 162-meter hallway that’s “aimed” at the Asinelli tower 1,407 meters from the window. This produces an odd effect: As you move north along the hallway toward the window, you’re approaching the tower, yet it seems to shrink. This is because the retinal size of the window’s aperture increases enormously as you approach it, while the retinal size of the distant tower remains relatively unchanged.

Similarly, as you back away from the window the tower seems to grow and draw closer, because the “shrinking” window shuts out the panorama, leaving only the tower in view. The illusion was first reported on a 1714 map by Paolo Battista Baldi of the University of Bologna.

The city contains a second “vista paradox” in the hermitage of Ronzano, where another long hallway is oriented toward the Sanctuary of San Luca 1,970 meters away. The Sanctuary seems to shrink as one approaches the frame and to grow as one retreats.

Pat Ashforth and Steve Plummer make knitted illusions. When it’s viewed from the front, each piece presents only a series of unremarkable stripes, but when it’s viewed obliquely, an image emerges.

Here’s an introduction to the technique, and here are some patterns.

Raymond Smullyan presented this oddity in his Chess Mysteries of Sherlock Holmes in 1980. Suppose we come upon this abandoned chess game. Can White mate in two moves? The answer seems to be yes. If Black can’t castle, then White can play 1. Ke6 and then promote his g-pawn, giving mate. If Black can castle, that means that neither his king nor his rook has yet moved, and hence he must just have moved his pawn to e5. That permits White to capture the pawn en passant. Now if Black castles then 2. b7 is mate, and if he plays any other move then the g-pawn promotes. Either way, White mates Black on his second move.

But that’s odd. We’ve decided that a mate in two exists, but we can’t show it — and we don’t even know how White commences!

(F. Alexander Norman, “Classicists and Constructivists: A Dilemma,” Mathematics Magazine 62:5 [December 1989], 340-342. See Donkey Sentences.)

02/17/2025 UPDATE: Reader Eugene Kruglov points out that 1. Ke6 works in Smullyan’s position whether or not Black castles — if he does, then 2. a8=Q is mate. The position below works as Smullyan intended — when Black castles, only the en passant capture leads to mate in two. (Thanks, Eugene.)