Seniority

Annie, Betty, Carrie, Darla, and Eve all have the same birthday, but all are different ages. On their shared birthday:

Darla said to Betty: “I’m 9 years older than Eve.”
Eve said to Betty: “I’m 7 years older than Annie.”
Annie said to Betty: “Your age is exactly 70 percent greater than mine.”
Betty said to Carrie: “Eve is younger than you.”
Carrie said to Darla: “The difference between our ages is 6 years.”
Carrie said to Annie: “I’m 10 years older than you.”
Carrie said to Annie: “Betty is younger than Darla.”
Betty said to Carrie: “The difference between your age and Darla’s is the same as the difference between Darla’s and Eve’s.”

Whenever one of them spoke to someone older than herself, everything she said was true, but when she spoke to someone younger, everything she said was false. How old is each person?

	Select Click for Answer
Annie – 30 Betty – 51 Carrie – 55 Darla – 46 Eve – 37 Carrie tells Annie she’s older than her by 10 years. If Carrie were really younger than Annie, this statement would be true, and that’s impossible, so Carrie must be older than Annie, just not by 10 years. Record that as a fact: Carrie is older than Annie (but not by 10 years). Carrie also lies to (younger) Annie that Betty is younger than Darla. Fact: Darla is younger than Betty. Darla tells the truth to (older) Betty that she’s 9 years older than Eve. Fact: Darla is 9 years older than Eve. Eve tells the truth to (older) Betty that she’s 7 years older than Annie. Fact: Eve is 7 years older than Annie. Annie tells the truth to (older) Betty that Betty’s age is 70 percent greater than her own. For Betty’s age to be a whole number, Annie’s age must be a multiple of 10. Since Betty is older than Darla, and Darla is 7 + 9 = 16 years older than Annie, that means Betty has to be more than 16 years older than Annie. The lowest multiple of 7 greater than 16 is 21. Fact: Annie is at least 30 years old (and definitely a multiple of 10). At this point, it appears that either Carrie or Betty is oldest and thus lying to everyone else. Let’s assume that Carrie is oldest and see whether that works. In that case, Carrie is lying to Darla that the difference in their ages is 6 years, but Betty is telling the truth to (older) Carrie that the difference between Carrie’s age and Darla’s is the same as the difference between Darla’s and Eve’s, namely 9 years. Let’s test this scenario, assuming Annie’s age is 30. Then we get, from youngest to oldest: Annie = 30, Eve = 37, Darla = 46, Betty = 51, Carrie = 55 Checking all statements and the age relations shows that this is an answer. Is it the only answer? If Annie’s age were 40, then Betty’s age would be 68, and Carrie’s age would be 65, so Carrie would not be the oldest, and that would be a fatal flaw. If Annie is older than 30, then Betty is older than Carrie, and Carrie is not the oldest. Hence the values above form the only consistent answer. From the Wikibooks puzzles collection (see that solution for a mathematical argument).

Click for Answer

Annie – 30
Betty – 51
Carrie – 55
Darla – 46
Eve – 37

Carrie tells Annie she’s older than her by 10 years. If Carrie were really younger than Annie, this statement would be true, and that’s impossible, so Carrie must be older than Annie, just not by 10 years.

Record that as a fact: Carrie is older than Annie (but not by 10 years).

Carrie also lies to (younger) Annie that Betty is younger than Darla.

Fact: Darla is younger than Betty.

Darla tells the truth to (older) Betty that she’s 9 years older than Eve.

Fact: Darla is 9 years older than Eve.

Eve tells the truth to (older) Betty that she’s 7 years older than Annie.

Fact: Eve is 7 years older than Annie.

Annie tells the truth to (older) Betty that Betty’s age is 70 percent greater than her own. For Betty’s age to be a whole number, Annie’s age must be a multiple of 10. Since Betty is older than Darla, and Darla is 7 + 9 = 16 years older than Annie, that means Betty has to be more than 16 years older than Annie. The lowest multiple of 7 greater than 16 is 21.

Fact: Annie is at least 30 years old (and definitely a multiple of 10).

At this point, it appears that either Carrie or Betty is oldest and thus lying to everyone else. Let’s assume that Carrie is oldest and see whether that works.

In that case, Carrie is lying to Darla that the difference in their ages is 6 years, but Betty is telling the truth to (older) Carrie that the difference between Carrie’s age and Darla’s is the same as the difference between Darla’s and Eve’s, namely 9 years. Let’s test this scenario, assuming Annie’s age is 30. Then we get, from youngest to oldest:

Annie = 30, Eve = 37, Darla = 46, Betty = 51, Carrie = 55

Checking all statements and the age relations shows that this is an answer. Is it the only answer?

If Annie’s age were 40, then Betty’s age would be 68, and Carrie’s age would be 65, so Carrie would not be the oldest, and that would be a fatal flaw. If Annie is older than 30, then Betty is older than Carrie, and Carrie is not the oldest. Hence the values above form the only consistent answer.

From the Wikibooks puzzles collection (see that solution for a mathematical argument).

May 12, 2026May 11, 2026 | Uncategorized

“To a Lost Sweetheart”

When Whistler’s Mother’s Picture’s frame
Split, that sad morn, in two,
Your tense words scorched me like a flame —
You shrieked, “Ah, glue! Get glue!”

O Glue! O God! there was not glue
Enough in all the feet
Of all the kine the wide world through
To hold you to me, Sweet!

— Don Marquis

May 11, 2026May 10, 2026 | Poems

A Union Cipher

This baffling message illustrates a cipher adopted by the Union Army in 1862:

TO GEORGE C. MAYNARD, Washington

Regulars ordered of my to public out suspending received 1862 spoiled thirty I dispatch command of continue of best otherwise worst Arabia my command discharge duty of my last for Lincoln September period your from sense shall duties the until Seward ability to the I a removal evening Adam herald tribune.

PHILIP BRUNER

The address and signature are “covers” that don’t enter into the cipher. The first word, Regulars, is a code indicating that the original message had been written in five columns of nine words each. Tribune, herald, spoiled, Seward, for, and worst are null words; Lincoln is code for Louisville, Kentucky; Adam means General Henry Wager Halleck; and Arabia is code for Major General Don Carlos Buell. The word Period indicates a full stop. This had been the original message:

Louisville, Kentucky
September thirty 1862

General Halleck:

(Adam)   (period)   I           received     last
evening  your       dispatch    suspending   my
removal  from       command.    Out          of
a        sense      of          public       duty,
I        shall      continue    to           discharge
the      duties     of          my           command
to       the        best        of           my
ability  until      otherwise   ordered.

D.C. Buell,
Major General

This message had been enciphered by reading up the fourth column, down the third, up the fifth, down the second, and up the first; inserting the null words; and encoding the most sensitive particulars. The system worked well until July 1864, when Union cipher operator Stephen L. Robinson was captured by Confederate guerrillas and the key seized.

(John Laffin, Codes and Ciphers Secret Writing Through the Ages, 1964.)

Conclusions

From John Boyce Bennett’s 1980 logic textbook Rational Thinking:

If it’s false that no dopips are fraks, characterize each of these propositions as true, false, or doubtful:

a. All dopips are fraks.
b. Few dopips are fraks.
c. Some dopips are fraks.
d. No fraks are dopips.
e. Some dopips are not fraks.

	Select Click for Answer
If it’s false that no dopips are fraks, then some dopips are fraks, so c is true. That in turn makes d false and the other statements doubtful.

May 10, 2026May 9, 2026 | Puzzles

The Last Straw

In Marriage and Morals (1929), Bertrand Russell mentions that, while “[c]ruelty is in theory a perfectly adequate ground for divorce, … it may be interpreted so as to become absurd”:

When the most eminent of all film stars was divorced by his wife for cruelty, one of the counts in the proof of cruelty was that he used to bring home friends who talked about Kant.

I haven’t been able to figure out who this is. Russell writes, “I hardly suppose that it was the intention of the California legislators to enable any woman to divorce her husband on the ground that he was sometimes guilty of intelligent conversation in her presence.”

05/11/2026 UPDATE: Russell was referring to the 1920 divorce of Mildred Harris from Charlie Chaplin. The two had married in 1918, when Chaplin was 29 and Harris was 16. In 2015, Silent Film Quarterly republished an interview with Harris: “He brought men home to dinner. But such men! Old, grave, and intellectual men! They were 50 years old or more. They talked of things I could not possibly understand. I was seventeen. What could I know of philosophy, or of Voltaire, or Rousseau, or Kant?” Thanks to everyone who wrote in about this.

May 9, 2026May 11, 2026 | Society

Black and White

puig y puig chess problem

By Esteban Puig y Puig, Skakbladet, 1906. White to mate in two moves.

	Select Click for Answer
White wins with the very surprising 1. Ra4! If Black takes the rook then the bishop will mate; if he tries to flee then the queen will finish the job.

May 8, 2026May 7, 2026 | Puzzles

Progress

I do not mean to be disrespectful, but the attempt of the lords to stop the progress of reform reminds me very forcibly of the great storm of Sidmouth, and of the conduct of the excellent Mrs. Partington on that occasion. In the winter of 1824, there set in a great flood upon that town; the tide rose to an incredible height; the waves rushed in upon the houses, and everything was threatened with destruction. In the midst of this sublime and terrible storm, Dame Partington, who lived upon the beach, was seen at the door of her house with mop and pattens, trundling her mop, squeezing out the sea-water, and vigorously pushing away the Atlantic Ocean. The Atlantic was roused. Mrs. Partington’s spirit was up; but I need not tell you that the contest was unequal. The Atlantic Ocean beat Mrs. Partington. She was excellent at a slop or a puddle, but she should not have meddled with a tempest. Gentlemen, be at your ease — be quiet and steady. You will beat Mrs. Partington.

— Sydney Smith on the Reform Bill, Taunton, Oct. 12, 1831

May 8, 2026May 7, 2026 | Society

The Brazil Nut Effect

https://commons.wikimedia.org/wiki/File:Mixed_nuts_small_white2.jpg — Image: Wikimedia Commons

When a container of granular material is shaken, we might expect the largest particles to make their way to the bottom. Instead the opposite often happens: Vibrating a container of mixed nuts, muesli, or raisin bran often brings the largest (and presumably heaviest) items to the top.

Precisely why this happens is unclear. An irregularly shaped Brazil nut might “shoulder” its way above smaller nuts as it turns among them; the rising of large particles might help to lower the center of mass of the aggregate; or perhaps the size of the largest particles prevents them from descending in a container’s natural convection flow once they reach the surface. For now it’s an unsolved problem in physics.

May 7, 2026May 6, 2026 | Science & Math

Couplet

It is remarkable that they
Talk most who have the least to say.

— Matthew Prior

May 7, 2026May 6, 2026 | Poems

Commitment

“Deddicacion” of Scottish philologist James Elphinstone’s 1786 proposal on spelling reform, Propriety Ascertained in Her Picture:

To’ dhe KING.

Sir,

Augustus found hiz Language ripe for immortallity: hiz smiles bade Roman Lerning ascertain Propriety in her Picture. A LEO’s goolden days gave rizing Tempels to’ ring widh hightened harmony; gave, not onely a Raphael to’ paint, but a Vida to sing. Reviving Art and Science danced down hand in hand. LEWIS, fostering Genius, and founding Accademies, rendered France dhe admiracion ov dhe World, and her Language dhe Diccion ov it. YOOR MADJESTY, emmulous no les ov preceding, dhan ov contemporary Glory; and finding Glory, onely in dhe improovment ov mankind; haz dained, not merely by pattronage ov dhe sublimest Muzic, and by dhe institucion ov a Brittish Acaddemy, to’ raiz rivals to’ dhe moast exquizite Artists ov Anticquity; but, by fixing Inglish Speech in Inglish Orthoggraphy, to’ secure dhe unfading luster ov Truith, and dhe unfailing succession ov a Horrace, a Boileau, and a Pope.

If an umbel individdual haz prezumed to’ attempt a task, hiddherto’ held arduous for Acaddemies; he hopes for pardon, onely az he shal be found to’ hav performed it: nor wil, in such case, dhe Smile be regretted, hwich constitutes him, widh so dutifool venneracion,

SIR,

YOOR MADJESTY’S

moast devotedly zellous,

az peculiarly onnored, Servant;

JAMES ELPHINSTONE.

May 6, 2026 | Language