# Not So Fast

José Paluzie offered this chess poser in 1910. White is to move and mate in 1:

The key is to notice that the position is illegal: There’s no legal way for the black king to have arrived at a2. Black, desperate to avoid mate, must have put it there when White wasn’t looking.

Where did it come from? It doesn’t matter: White can place the black king on any legal square and mate in 1.

(From Burt Hochberg’s Chess Braintwisters, 1999.)

# Balance

Point P lies within acute angle XOY. How can we find a point A on OX and a point B on OY such that P is the midpoint of a segment drawn between them?

# A Rolling Lemon

My lousy car has an odometer without 4s — in every position, the counter advances from 3 directly to 5. For example, when it read 000039 I drove one mile and watched it roll over to 000050. Today the odometer reads 002005. How many miles has the car actually traveled?

# Left or Right?

You come upon the track of a bicycle in the mud. Was the bicycle traveling to the left or the right?

# Black and White

By Paul Helweg Mikkelsen. White to mate in two.

# Pagan Island

Twenty-six villages are ranged around the coastline of an island. Their names, in order, are A, B, C, …, Z. At various times in its history, the island has been visited by 26 missionaries, who names are also A, B, C, …, Z. Each missionary landed first at the village that bore his name and began his work there. Each village was pagan to begin with but became converted when visited by a missionary. Whenever a missionary converted a village he would move along the coastline to the next village in the cycle ABC-…-ZA. If a missionary arrived at an uncoverted village he’d convert it and continue along the cycle, but there was never more than one missionary in a village at a time. If a missionary arrived at a village that had already been converted, the villagers, feeling oppressed, would kill him and revert to a state of paganism; they would do this even to a missionary who had converted them himself and then traveled all the way around the island. There’s no restriction as to how many missionaries can be on the island at any given time. After all 26 missionaries have come and gone, how many villages remain converted?

# Black and White

By Oskar Blumenthal. White to mate in two moves.

In 1894 Lewis Carroll published a conundrum that, he wrote, presents “a very real difficulty in the Theory of Hypotheticals.” Suppose that Allen, Brown, and Carr run a shop. At least one of them must always be present to mind the shop, and whenever Allen leaves he always takes Brown with him. Now, suppose that Carr is out. In that case then if Allen is out then Brown must be in, in order to tend the shop. But we know that this isn’t true — we’ve been told that whenever Allen is out then Brown is out.

Since the supposition that Carr is out leads to a falsehood, then it must itself be false. Confusingly, the laws of logic seem to require that Carr never leave the shop.

“I greatly hope that some of the readers of Mind who take an interest in logic will assist in clearing up these curious difficulties,” Carroll wrote. Modern logicians would say that this is a simple error in reasoning, rather than a logical disaster. But what is the error?

# The Falling Chain

Here are two identical rope ladders with slanting rungs. One falls to the floor, the other onto a table. The ladders are released at the same time and fall freely, but the one on the left falls faster, as if the table is “sucking” it downward. Why does this happen?

# Podcast Episode 100: Lateral Thinking Puzzles

Here are five new lateral thinking puzzles to test your wits and stump your friends — play along with us as we try to untangle some perplexing situations using yes-or-no questions.

See full show notes …