Litton's Problematical Recreations

Practice flashcards based on the mathematical puzzles and recreations in Litton's Problematical Recreations.

In Litton's Problematical Recreations, Rufus T. Flypaper drives $2\,\text{miles}$ to work each morning at an average target speed of $30\,\text{mph}$. If a driver slows him down for the first mile so his average speed is $15\,\text{mph}$, can he still arrive on time assuming his car can do $120\,\text{mph}$?

No

A hunter's one-piece rifle has a length of $1.7\,\text{yards}$. A baggage man refuses any article whose greatest dimension exceeds $1\,\text{yard}$. How can the hunter pack the rifle so it is permitted?

Put it diagonally in a cubical box $1\,\text{yard}$ on a side.

Two trains leave stations A and B $120\,\text{miles}$ apart, traveling toward each other at $25\,\text{mph}$ and $15\,\text{mph}$ respectively. A South American botfly flies back and forth between the engines at $100\,\text{mph}$ until the collision. How far does the fly travel?

$$300\,\text{miles}$$

How can a bricklayer find the heaviest brick among $8$ bricks (where $7$ weigh the same and $1$ is heavier) using a balance scale in only $2$ weighings?

Divide the bricks into groups of $3$, $3$, and $2$. Weigh two groups of $3$ against each other first.

Maynard's grandfather clock has weights for the time and striking mechanisms that are exactly opposite each other after winding at bedtime, and opposite every $6\,\text{hours}$ thereafter. What is Maynard's bedtime?

$9\,\text{pm}$ or $3\,\text{am}$

Dr. Fubisher LaRouche found that $1$ item costs $10\,\text{cents}$, $8$ costs $10\,\text{cents}$, $17$ costs $20\,\text{cents}$, $104$ costs $30\,\text{cents}$, $756$ costs $30\,\text{cents}$, and $1072$ costs $40\,\text{cents}$. What was he buying?

House numbers ($10\,\text{cents}$ per digit/number).

How can one construct $1000$ consecutive nonprime numbers?

Use the sequence $! + 2, 1001! + 3, \dots, 1001! + 1001$.

In a complete $9$-inning Major League baseball game, what is the minimum number of pitches pitcher Hi N. Outside could make?

$$25\,\text{pitches}$$

How can it be demonstrated that an irrational number raised to an irrational power can be rational?

Consider $A = \sqrt{2}^{\sqrt{2}}$. If $A$ is rational, it serves as the example; if $A$ is irrational, then $A^{\sqrt{2}} = (\sqrt{2}^{\sqrt{2}})^{\sqrt{2}} = \sqrt{2}^2 = 2$ is rational.

What limerick translates the equation $\frac{12 + 144 + 20 + 3\sqrt{4}}{7} + 5(11) = 9^2 + 0$?

A dozen, a gross and a score / Plus three times the square root of four, / Divided by seven, / Plus five times eleven, / Is nine squared and not a bit more.

What are two ways of representing $20$ using three $3$'s and standard mathematical symbols?

$\frac{(3!)!}{(3!)(3!)}$ or $\left(\frac{3 + 3}{3}\right)!$

In the yacht problem involving Mary Ann Moore's father and his four friends (Colonel Downing, Mr. Hall, Sir Barnacle Hood, Dr. Parker), who is Lorna's father?

Colonel Downing

Three blindfolded men with marked foreheads raise their hands if they see a mark. When blindfolds are removed, all hands raise. Shortly, one man lowers his hand. What is his logical reasoning?

He reasons that if his forehead were unmarked, the other two would see only one mark each and immediately realize their own foreheads were marked and lower their hands. Since neither did, his own forehead must also be marked.

What mathematical property is common to Arctic penguins, peacock eggs, the Hungarian Merchant Marine, the University of Chicago football team, $19$ point cribbage hands, and the solution set of $e^{e^x} = 1$?

Each class is empty.

In Bristol, $90\%$ drink tea, $80\%$ drink coffee, $70\%$ drink whiskey, and $60\%$ drink gin, with no one drinking all four. What percentage of Bristol's citizens drink liquor?

$$100\%$$

How should a prisoner arrange $10$ white balls and $10$ black balls into two boxes to maximize his chance of drawing a white ball and surviving?

Place $1$ white ball in one box, and the remaining $19$ balls ($9$ white and $10$ black) in the other box, yielding a survival probability of $73.7\%$.

In a duel between two $50\%$ marksmen who alternate shots until one is hit, what are the odds in favor of the shooter who goes first?

$$\frac{2}{3}$$

For $n$ points on a circle with no $3$ collinear chord intersections, what is the formula for the number of internal intersections?

$$\frac{n(n-1)(n-2)(n-3)}{4 \cdot 3 \cdot 2 \cdot 1}$$

Two marbles removed from a bag of black and white marbles have a $\frac{1}{3}$ chance of both being white. If three are removed, the chance is $\frac{1}{6}$. How many marbles of each color are in the bag?

$6$ white marbles and $4$ black marbles.

In an unbalanced coin, getting two heads in two successive throws is as likely as getting tails in one throw. What is the probability of getting heads in a single throw?

$$0.618$$

A math professor pays his son $8 for every correct partial differential equation solved and fines him $5 for every incorrect solution. After $26$ problems, neither owes money. How many did the son solve correctly?

$$10$$

For $X < 1$, evaluate the infinite product $(1 + X + X^2 + \dots + X^9)(1 + X^{10} + X^{20} + \dots + X^{90})(1 + X^{100} + X^{200} + \dots + X^{900})\dots$

$$\frac{1}{1-X}$$

Dr. Reed notices his watch hands are exactly together every $65$ minutes. Does his watch gain or lose, and by how much per hour?

Gains $\frac{60}{143}\,\text{minutes}$ per hour.

Dr. Irving Weiman walks up an upgoing escalator at $1\,\text{step/sec}$ and reaches the top in $20$ steps. At $2\,\text{steps/sec}$, he reaches the top in $32$ steps. How many steps are on the escalator?

$$80\,\text{steps}$$

Why did Archimedes O'Toole note without calculation that the sum and product of $\frac{x+y-|x-y|}{2}$ and $\frac{x+y+|x-y|}{2}$ are $x+y$ and $xy$?

The two expressions are identically equal to the smaller and larger of the two numbers $x$ and $y$, respectively.

Solve the equation $\sqrt{x+\sqrt{x+\sqrt{x+\dots}}} = \sqrt{x\sqrt{x\sqrt{x\dots}}}$ for real values of $x$.

$x = 0$ or $x = 2$

Solve for real values of $x$: $(7+4\sqrt{3})^x - 4(2+\sqrt{3})^x = -1$.

$$x = 1$$

A spider and a fly are at opposite vertices of a rectangular room with dimensions $1$, $2$, and $3$ units. What is the minimum crawling distance for the spider to reach the fly?

$\sqrt{18}$ units

In a room $40\,\text{feet}$ long, $20\,\text{feet}$ wide, and $20\,\text{feet}$ high, a bug crawls along the walls from a point $1\,\text{foot}$ above the floor on one end wall to a point $1\,\text{foot}$ below the ceiling on the opposite end wall. What is the shortest route?

$$58\,\text{ft}$$

What three distinct visual configurations can be perceived in the optical illusion shown in Problem 37 of Chapter 5?

1. A little cube nestled in the corner of a big one. 2. A big cube with a cubical chunk removed from one corner. 3. Two cubes meeting externally at a corner.

What is the simplest integer solution for the equation $\frac{1}{x^2} + \frac{1}{y^2} = \frac{1}{z^2}$?

$x = 15$, $y = 20$, $z = 12$

The product of three occupants' ages is $1296$ and their sum is their house number. After learning that the owner's son and grandson live with him, the census taker finds their ages. What are the ages and street number?

Ages are $1$, $18$, and $72$; street number is $72$.

In Byzantine basketball, $35$ score totals are impossible, including $58$. What are the point values for a free throw and a field goal?

Free throw = $8$ points, field goal = $11$ points.

If $\text{THAT} = (\text{AH})(\text{HA})$, what four-digit integer is represented by $\text{THAT}$?

$$6786$$

What is the smallest number of parade marchers such that marching $3$ abreast leaves $2$ over, $5$ abreast leaves $4$ over, $7$ abreast leaves $6$ over, and $11$ abreast leaves $10$ over?

$$1154$$

What is the remainder when dividing $5^{999,999}$ by $7$?

$$6$$

In the arithmetic of Puevigi, $14$ is a factor of $41$. What is the base of the number system?

$$11$$

Which value is greater: $e^{\pi}$ or $\pi^e$?

$$e^{\pi}$$

What is the sum of the infinite series $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \frac{1}{12} + \dots$ whose terms are reciprocals of positive integers divisible by no prime greater than $3$?

$$3$$

What limerick translates the integral equation $\int_1^{\sqrt[3]{3}} z^2\,dz \times \cos\left(\frac{3\pi}{9}\right) = \ln(\sqrt[3]{e})$?

Integral z squared dz / from one to the cube root of three / Multiplied by cosine / of three pi over nine / is the log of the cube root of e.