Book:David Wells/Curious and Interesting Puzzles/Errata

Errata for 1992: David Wells: Curious and Interesting Puzzles

Rational Number Expressible as Sum of Reciprocals of Distinct Squares

The Puzzles: Egyptian Fractions

The sum of the series $1 + 1 / 2^2 + 1 / 3^2 + 1 / 4^2 \ldots = \pi^2 / 6$, so the sum of different Egyptian fractions whose denominators are squares cannot exceed $\pi^2 / 6$, but might equal, for example, $\frac 1 2$.

Think of a Number: Rhind Papyrus $28$

The Puzzles: Think of a Number

Problem $29$ of the Rhind papyrus is not quite so clear, but it is plausibly the first ever 'Think of a Number' problem. It reads ...

Division of Triangular Field

The Puzzles: Dividing a Field: $12$

A triangular field is to be divided between six brothers by equidistant lines parallel to one side. The length of the marked side is $6, 30$ and the area is $11, 22, 30$. What is the difference between the brothers' shares?

Answer: The difference between each successive share is $37, 55$ or $37 \frac {11} {12}$.

Greek Anthology Book $\text {XIV}$: $7$

The Puzzles: Light Reflected off a Mirror: $19$

My right eye fills $1/8$ jar in $6$ hours (taking a day to be $24$ hours, where the Greeks might have taken it to be $12$), and my left eye fills $1/12$ in $6$ hours, and my foot $1/16$. Thus all four fill the jar $1 + 1/8 + 1/12 + 1/16 = \frac {13} {48}$ times in $6$ hours. So the jar will be filled once in $6 \times 48/61$ hours, or $47$ minutes and $13$ seconds, approximately.

(Sandford, $1930$, p. $216$)

Workman Example

The Puzzles: Light Reflected off a Mirror: $20$

$A$ and $B$ together can do a piece of work in $6$ days, $B$ and $C$ together in $20$ days, $C$ and $A$ together in $7 \frac 1 2$ days. How long will each require separately to do the same work?

The solution is given as:

$A$, $B$ and $C$ do the whole job in $10$, $15$ and $30$ days respectively.

Inscribing Equilateral Triangle inside Square with a Coincident Vertex

The Puzzles: Abul Wafa ($\text {940}$ – $\text {998}$): $38$

Abul Wafa gave five different solutions. Here are three of them. ...

... Join $B$ to the midpoint, $M$, of $DC$. Draw an arc with centre $B$ and radius $BA$ to cut $BM$ at $N$. Let $DN$ cut $CB$ at $H$. Then $H$ is one of the vertices sought.

Dissection of 2 Regular Hexagons into One

The Puzzles: Abul Wafa ($\text {940}$ – $\text {998}$): $41$

How can two regular hexagons, of different sizes, be dissected into seven pieces which fit together to make one, larger, regular hexagon?

Note on Xugu Zhaiqi Suanfa

The Puzzles: Sun Tsu Suan-Ching

Yang Hui (c. $1270$ ad) wrote an 'Arithmetic in Nine Sections', which contains the very first extant representation of what we in the West call Pascal's Triangle (from an earlier Chinese source, c. $1000$ ad). His book was called, apparently, Hsu Ku Chai Chi Suan Fa ($1275$). It contains the following magic configuration: ...

Josephus Problem: Turks and Christians Variant

The Puzzles: The Josephus Problem: $99$

On board a ship, tossed in storms and in danger of shipwreck, are fifteen Christians and fifteen Turks. To lighten the load and save the ship, half are to be thrown overboard. One of the Christians suggests that all should stand in a circle and every ninth person counting around the circle should be chosen. How should the Christians arrange themselves in the circle to ensure that only the Turks die?

The solution is given as:

The Christians and Turks should be placed in the following circular order, in which the first person follows the last, and the counting starts with the first person: $\text {CCCCTTTTTCCTCCCTCTTCCTTTCTTTCCTT}$.

Proper Integer Heronian Triangle whose Area is $24$

The Puzzles: Bachet: $111$

What is the unique Heronian triangle with area $24$?

Arrange 3 Knives to Rest on Tips of Handles: Variant

The Puzzles: Henry van Etten: $112$

Variants in Victorian puzzle books demanded how three knives might be used to support a drinking glass, in the ample space between three other drinking glasses placed on the table with more than enough space for a fourth glass to be placed on the table between them.

Balance a Stick on its End on a Finger

The Puzzles: Henry van Etten: $113$

How can a stick be made to balance securely on the tip of a finger?

The solution is given as:

Force the tips of three knives into the stick, so that the knives hang well below the finger.

The centre of gravity of the entire arrangement will then be below the fingertip and will be stable.

Trefoil Knot in Paper forms Pentagon

The Puzzles: Henry van Etten: $126$

The next problem occurs first in Urbino d'Aviso's treatise on the sphere (1682):

Relative Likelihood of n Sixes on 6n Dice

The Puzzles: Prince Rupert's Cube: $129$

Our short table shows the probabilities, rounded to three decimals, of obtaining the mean number or more sixes when $6 n$ dice are tossed.

$6 n$	$n$	$\map P {\text {$n$ or more sixes} }$
$6$	$1$	$0.665$
$12$	$2$	$0.619$
$18$	$3$	$0.597$
$24$	$2$	$0.584$
$30$	$5$	$0.576$
$96$	$6$	$0.542$
$600$	$100$	$0.517$
$900$	$150$	$0.514$

Number of Derangements on Finite Set

The Puzzles: The misaddressed letters: $130$

A correspondent writes ten letters and addresses ten envelopes, one for each letter. In how many ways can all the letters be placed in the wrong envelopes?

The solution given is:

Bernoulli posed the problem in terms of $n$ letters wrongly placed into $n$ envelopes, but the principle is the same. ...

When $n = 7$, the value is $1854$.

Orthogonal Latin Squares of Order $5$

The Puzzles: The Thirty-six Officers Problem: $135$

How can five each of As, Bs, Cs, Ds and Es be placed in these cells so that no letter is repeated in any row or column?

$\begin{array} {|c|c|c|c|c|} \hline \ \ & \ \ & \ \ & \ \ & \ \ \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & \\ \hline \end{array}$

Rational Amusements for Winter Evenings

The Puzzles: Rational Amusements for Winter Evenings

John Jackson was 'A Private Teacher of Mathematics' who decided that there were many puzzles scattered around, but not collected together in one small and convenient volume, so he assembled them himself and wrote Rational Amusements for Winter Evenings, or, A Collection of above 200 Curious and Interesting Puzzles and Paradoxes relating to Arithmetic, Geometry, Geography, &, 'Designed Chiefly for Young Persons' , which appeared in London in $1821$.