Squaring Words

In his 1864 autobiography Passages From the Life of a Philosopher, Charles Babbage describes an “amusing puzzle.” The task is to write a given word in the first rank and file of a square and then fill the remaining blanks with letters so that the same four words appear in order both horizontally and vertically. He gives this example with the word DEAN:


“The various ranks of the church are easily squared,” he writes, “but it is stated, I know not on what authority, that no one has yet succeeded in squaring the word bishop.”

By an unlikely coincidence I’ve just found that Eureka put this problem to its readers in 1961, and they found three solutions:

B I S H O P    B I S H O P    B I S H O P
I L L U M E    I N H E R E    I M P A L E
S L I D E S    S H A R P S    S P I N E T
H U D D L E    H E R M I T    H A N G A R
O M E L E T    O R P I N E    O L E A T E
P E S E T A    P E S T E R    P E T R E L

The first was found by A.L. Cooil and J.M. Dagnese; the second by A.R.B. Thomas; and the third by R.W. Payne, J.D.E. Konhauser, and M. Rumney.

12/10/2023 UPDATE: Reader Giorgos Kalogeropoulos has enlisted a database of 235,000 words to produce more than 100 bishop squares (click to enlarge):

Kalogeropoulos bishop squares

This is pleasing, because it’s a road that Babbage himself was trying to follow in the 19th century, laboriously cataloging the contents of physical dictionaries after an algorithm of his own devising — see page 238 in the book linked above. (Thanks, Giorgos.)

A Puzzle Forest

An unusual problem by Reddit user cgibbard, from a discussion in 2010:

Here’s a representation of 101010 for you to figure out.

      *             *      *
      |             |      |
   *  *  *   *  *   *  * * *
   |  |   \ /    \ /    \|/
*  *  *    *      *      *

As a bonus, here’s the corresponding representation for 42:

   * *   *
   |  \ /
*  *   *

The puzzle is to find the rule for this representation.

A commenter wrote, “This puzzle is really clever and very rewarding to figure out.”

Click for Answer



“Here is an old favorite of mine that completely stumped me once when I was a student,” writes Jeff Hooper in Crux Mathematicorum. Suppose that each square in this 3×7 grid is painted red or black at random. Show that the board must contain a rectangle whose four corner squares are all the same color.

Click for Answer



A problem by Russian mathematician Viktor Prasolov: Prove that it’s impossible to cut a 10×10 chessboard into T-shaped tiles of 4 squares each.

Click for Answer


A problem from the Stanford University Competitive Examination in Mathematics:

How old is the captain, how many children has he, and how long is his boat? Given the product 32118 of the three desired numbers (integers). The length of the boat is given in feet (is several feet), the captain has both sons and daughters, he has more years than children, but he is not yet one hundred years old.

Click for Answer


From John Scott, The Puzzle King, 1899:

“A farmer, being asked what number of animals he kept, answered: ‘They’re all horses but two, all sheep but two, and all pigs but two.’ How many had he?”

Click for Answer

A Self-Descriptive Crossword Puzzle

From Lee Sallows:

Can you complete the ‘self-descriptive crossword puzzle’ at left below? As in the solution to a similar puzzle seen at right, each of its 13 entries, 6 horizontal, 7 vertical, consists of an English number name folowed by a space followed by a distinct letter. The number preceding each letter describes the total number of occurrences of the letter in the completed puzzle. Hence, in the example, E occurs thirteen times, G only once, and so on, as readers can check. Note that the self-description is complete; every distinct letter is counted.

Though far from easy, the self-descriptive property of the crossword enables its solution to be inferred from its empty grid using reasoning based on orthography only.

sallows self-descriptive crossword

Click for Answer

Snow Manipulation


A puzzle by James M., an operations researcher at the National Security Agency:

Frosty the Snowman wants to create a small snowman friend for himself. The new snowman needs a base, torso, and a head, all three of which should be spheres. The torso should be no larger than the base and the head should be no larger than the torso.

For building material, Frosty has a spherical snowball with a 6-inch radius. Since Frosty likes to keep things simple, he also wants the radius of each of the three pieces to be a positive integer. Can Frosty accomplish this?

Click for Answer

“The Unlucky Hatter”

From The Book of 500 Curious Puzzles, 1859:

A blackleg passing through a town in Ohio, bought a hat for $8 and gave in payment a $50 bill. The hatter called on a merchant near by, who changed the note for him, and the blackleg having received his $42 change went his way. The next day the merchant discovered the note to be a counterfeit, and called upon the hatter, who was compelled forthwith to borrow $50 of another friend to redeem it with; but on turning to search for the blackleg he had left town, so that the note was useless on the hatter’s hands. The question is, what did he lose — was it $50 besides the hat, or was it $50 including the hat?

This is not so much a puzzle as a perplexity. “[I]n almost every case the first impression is, that the hatter lost $50 besides the hat, though it is evident he was paid for the hat, and had he kept the $8 he needed only to have borrowed $42 additional to redeem the note.”