The Three Cups Problem

Here are three cups, one upside down.

Turning over exactly two cups with each move, can you turn all cups right-side-up in no more than six moves?

If it’s possible, show how; if it’s not, say why.

“The Staircase Race”

A puzzle from Henry Dudeney’s Modern Puzzles and How to Solve Them, 1926:

This is a rough sketch of the finish of a race up a staircase in which three men took part. Ackworth, who is leading, went up three risers at a time, as arranged; Barnden, the second man, went four risers at a time, and Croft, who is last, went five at a time.

Undoubtedly Ackworth wins. But the point is, How many risers are there in the stairs, counting the top landing as a riser?

I have only shown the top of the stairs. There may be scores, or hundreds, of risers below the line. It was not necessary to draw them, as I only wanted to show the finish. But it is possible to tell from the evidence the fewest possible risers in that staircase. Can you do it?

Elementary

A logic exercise by Lewis Carroll: What conclusion can be drawn from these premises?

1. All the human race, except my footmen, have a certain amount of common sense.
2. No one who lives on barley sugar can be anything but a mere baby.
3. None but a hopscotch player knows what real happiness is.
4. No mere baby has a grain of common sense.
5. No engine driver ever plays hopscotch.
6. No footman of mine is ignorant of what true happiness is.

Real Estate

A curious problem from the Stanford University Competitive Examination in Mathematics: Bob wants a piece of land that’s exactly level and has four boundary lines, two running precisely north-south and two precisely east-west. And he wants each boundary line to measure exactly 100 feet. Can he buy such a piece of land in the United States?

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:

```D E A N
E A S E
A S K S
N E S T
```

“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):

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.”

Cornered

“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.

Cutup

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.

Profile

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.