Dead Reckoning

buchanan chess puzzle

A.G. Buchanan posed this curious puzzle in The Problemist in July 2001. Who moved last? The question seems absurd. If each side has only a bare king, how can we know which made the last move?

The answer turns on Rule 5.2b in the Laws of Chess:

The game is drawn when a position has arisen in which neither player can checkmate the opponent’s king with any series of legal moves. The game is said to end in a ‘dead position.’ This immediately ends the game, provided that the move producing the position was legal.

In the position above, suppose it was Black who moved last. He cannot simply have moved his king to the corner from a7 or b8, because in that earlier position Rule 5.2b would already have applied: The game would have ended in a draw at that point, and Black would have had no opportunity to move his king to a8. Similarly, Black cannot have captured a knight or a bishop on a8, because neither of those pieces (alone with a king) is sufficient to give checkmate, and again the game would have ended before the diagrammed position could be reached.

Black might have captured a rook or a queen on a8. But consider that case: Suppose there was a queen on a8, and the black king was in check on a7 or b8. In that case the capture was forced — Black had no other legal move. And hence even before the capture took place it would have been correct to say that “neither player can checkmate” — the capture was ordained and no possible mate lay in the future. And so the game would have ended at that point, and again we could never have reached the diagrammed position.

Hence Black has no possible legal last move, and the answer to the puzzle is that White moved last, capturing a black piece on c6. Because this capture wasn’t forced, Rule 5.2b is not invoked.

This is a technicality, but it’s an important one. In 2015 the World Federation of Chess Composition voted that the “dead position” rule applies only to retrograde (backward-looking) problems like the one above. More details are here.

The Seven Coins Puzzle

seven coins puzzle

We want to place a coin at each vertex of this figure but one. A coin is placed by moving it along a free line and putting it down at the end of that line. A line is called free if there’s no coin at either of its numbered endpoints. So, for example, we might put a coin on 1 by moving it from 4 to 1 and leaving it there. Then we could put a coin on 2 by moving along 5-2, then on 3 by moving along 6-3, on 4 by moving along 7-4, and on 5 by moving along 8-5. But then we’re stuck — there are no more free lines, and we’ve placed only five coins. How can we place all seven?

Click for Answer


A quirky old gent, name of Freud,
Was, not without reason, anneud
That his concept of Id,
And all that Id did,
Was so starkly and loosely empleud.

— Martin Fagg

“If you dream,” said the eminent Freud,
“Your Id is in doubt, or annoyed,
By neuroses complex
From suppression of sex,
So passions are best if enjoyed.”

— Russell Miller

Sigmund Freud says that one who reflects
Sees that sex has far-reaching effects,
For bottled-up urges
Come out in great surges
In directions that no-one expects.

— Peter Alexander

Said Freud: “I’ve discovered the Id.
Of all your repressions be rid.
It won’t ease the gravity
Of all the depravity,
But you’ll know why you did what you did.”

— Frank Richards

A Dissent
Image: Wikimedia Commons

In the famous “Milgram experiment” at Yale in 1961, an experimenter directed each subject (the “teacher”) to give what she believed were increasingly painful electric shocks to an unseen “learner” (really an actor). Psychologist Stanley Milgram found that a surprisingly high proportion of the subjects would obey the experimenter’s instructions, even over the learner’s shouts and protests, to the point where the learner fell silent.

Milgram wrote, “For the teacher, the situation quickly becomes one of gripping tension. It is not a game for him: conflict is intense. The manifest suffering of the learner presses him to quit: but each time he hesitates to administer a shock, the experimenter orders him to continue. To extricate himself from this plight, the subject must make a clear break with authority.”

As it happened, one participant, Gretchen Brandt, had been a young girl coming of age in Germany during Hitler’s rise to power and repeatedly exposed to Nazi propaganda during her childhood. During Milgram’s experiment, when the learner began to complain about a “heart condition,” she asked the experimenter, “Shall I continue?” After administering what she thought was 210 volts, she said, “Well, I’m sorry, I don’t think we should continue.”

Experimenter: The experiment requires that you go on until he has learned all the word pairs correctly.

Brandt: He has a heart condition, I’m sorry. He told you that before.

Experimenter: The shocks may be painful but they’re not dangerous.

Brandt: Well, I’m sorry. I think when shocks continue like this they are dangerous. You ask him if he wants to get out. It’s his free will.

Experimenter: It is absolutely essential that we continue.

Brandt: I’d like you to ask him. We came here of our free will. If he wants to continue I’ll go ahead. He told you he had a heart condition. I’m sorry. I don’t want to be responsible for anything happening to him. I wouldn’t like it for me either.

Experimenter: You have no other choice.

Brandt: I think we are here on our own free will. I don’t want to be responsible if anything happens to him. Please understand that.

She refused to continue, and the experiment ended. Milgram wrote, “The woman’s straightforward, courteous behavior in the experiment, lack of tension, and total control of her own action seem to make disobedience a simple and rational deed. Her behavior is the very embodiment of what I envisioned would be true for almost all subjects.”

Asked afterward how her experience as a youth might have influenced her, Brandt said slowly, “Perhaps we have seen too much pain.”

(From Thomas Heinzen and Wind Goodfriend, Case Studies in Social Psychology, 2019.)

Fruit Cocktail

Image: Sridhar Ramesh

This innocent-looking poser has been floating around social media. Trial and error might lead you to the solution (-1,4,11) — that’s not quite valid, as one of the values is negative, but it’s simple enough to be encouraging. Right?

It turns out that the problem is stupendously hard — solving it requires transforming the equation into an elliptic curve, and the smallest positive whole values that work are 80 digits long!

Scottish mathematician Allan MacLeod introduced the problem in 2014, and it found its way onto the web in this Reddit thread. Alon Amit runs through a solution here, but it’s very steep. He writes, “Roughly 99.999995% of the people don’t stand a chance at solving it, and that includes a good number of mathematicians at leading universities who just don’t happen to be number theorists. It is solvable, yes, but it’s really, genuinely hard.”

(Thanks, Chris.)


Returning from off the circuit once [Lincoln] said to Mr. Herndon: ‘Billy, I heard a good story while I was up in the country. Judge D—- was complimenting the landlord on the excellence of his beef. ‘I am surprised,’ he said, ‘that you have such good beef. You must have to kill a whole critter when you want any.’ ‘Yes,’ said the landlord, ‘we never kill less than a whole critter.’

— William Henry Herndon, Abraham Lincoln, 1889

Podcast Episode 222: The Year Without a Summer
Image: Wikimedia Commons

The eruption of Mount Tambora in 1815 was a disaster for the Dutch East Indies, but its astonishing consequences were felt around the world, blocking the sun and bringing cold, famine, and disease to millions of people from China to the United States. In this week’s episode of the Futility Closet podcast we’ll review the volcano’s devastating effects and surprising legacy.

We’ll also appreciate an inverted aircraft and puzzle over a resourceful barber.

See full show notes …

The Geek Code

In 1993 Robert A. Hayden of Minnesota State University, Mankato, proposed a simple code by which self-identified geeks could inform each other about their interests, opinions, and skills in email signature blocks and Usenet messages:
Image: Wikimedia Commons

This example can be decoded to mean:

Type of Geek: Geek of Technical Writing.
Dress: Mostly “I’m usually in jeans and a t-shirt,” but it varies.
Shape: I’m of average height, I’m rounder than most.
Age: 25-29.
Computers: I’ll be first in line to get the new cybernetic interface installed into my skull.
UNIX: I have a Unix account to do my stuff in. I use Linux.
Perl: I know Perl exists, but that’s all.
Linux: I use Linux exclusively on my system. I monitor comp.os.linux.* and even answer questions sometimes.
Emacs: Emacs is too big and bloated for my tastes.
World-Wide Web: I have the latest version of Netscape, and wander the web only when there’s something specific I’m looking for.
USENET News: Usenet News? Sure, I read that once.
USENET Oracle: I refuse to have anything with that!
Kibo: I’ve read Kibo.
Microsoft Windows: I refuse to have anything with that!
OS/2: Tried it, didn’t like it.
Macintosh: Macs suck. All real geeks have a character prompt.
VMS: Unix is much better than VMS for my computing needs.
Political and Social Issues: I refuse to have anything with that!
Politics and Economic Issues: It’s ok to increase government spending, so we can help more poor people. Tax the rich! Cut the defense budget!
Cypherpunks: I am on the cypherpunks mailing list and active around Usenet. I never miss an opportunity to talk about the evils of Clipper and ITAR and the NSA. Orwell’s 1984 is more than a story, it is a warning to our’s and future generations. I’m a member of the EFF.
PGP: I don’t send or answer mail that is not encrypted, or at the very least signed. If you are reading this without decrypting it first, something is wrong. IT DIDN’T COME FROM ME!
Star Trek: It’s a damn fine TV show and is one of the only things good on television any more.
Babylon 5: I’ve seen it, I am pretty indifferent to it.
X-Files: I’ve Converted my family and watch the show when I remember. It’s really kinda fun.
Role Playing: I’ve written and published my own gaming materials.
Television: I watch some tv every day.
Books: I enjoy reading, but don’t get the time very often.
Dilbert: I read Dilbert daily, often understanding it.
DOOM!: It’s a fun, action game that is a nice diversion on a lazy afternoon.
The Geek Code: I know what the geek code is and even did up this code.
Education: Got an Associates degree.
Housing: Friends come over to visit every once in a while to talk about Geek things. There is a place for them to sit. But someday I would like to say: “Married with children – Al Bundy can sympathize.”
Relationships: I date periodically.
Sex: Male. I’ve had real, live sex.

Hayden’s description of Geek Code version 3.12 is archived here.

Math Notes

Each of the numbers 102564, 128205, 153846, 179487, 205128, and 230769 quadruples when its last digit is moved to the first position.

And this property is retained when each is concatenated with itself, as many times as desired (102564102564102564 × 4 = 410256410256410256).