Henry Ernest Dudeney/Modern Puzzles/Arithmetical and Algebraical Problems/Locomotion and Speed Puzzles

Henry Ernest Dudeney: Modern Puzzles: Arithmetical and Algebraical Problems: Locomotion and Speed Puzzles

$28$ - Hill Climbing

Weary Willie went up a certain hill at the rate of $1 \tfrac 1 2$ miles per hour

and came down at the rate of $4 \tfrac 1 2$ miles per hour,

so that it took him just $6$ hours to make the double journey.

Now, how far was it to the top of the hill?

$29$ - Timing the Motor-car

"I was walking along the road at $3 \tfrac 1 2$ miles an hour," said Mr. Pipkins,

"when the motor-car dashed past me and only missed me by a few inches."

"Do you know what speed it was going?" asked his friend.

"Well, from the moment it passed me to its disappearance round a corner I took $27$ steps, and walking on reached that corner with $135$ steps more."

"Then, assuming you walked, and the car ran, each at a uniform rate, we can easily work out the speed."

$30$ - The Staircase Race

This is a rough sketch of a race up a staircase in which $3$ men took part.

Ackworth, who is leading, went up $3$ risers at a time, as arranged;

Barnden, the second man, went $4$ risers at a time,

and Croft, who is last, went $5$ 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?

$31$ - A Walking Puzzle

A man set out at noon to walk from Appleminster to Boneyham,

and a friend of his started at $2$ p.m. on the same day to walk from Boneyham to Appleminster.

They met on the road at $5$ minutes past $4$ o'clock

and each man reached his destination at exactly the same time.

Can you say what time they both arrived?

$32$ - Riding in the Wind

A man on a bicycle rode a mile in $3$ minutes with the wind at his back,

but it took him $4$ minutes to return against the wind.

How long would it take him to ride a mile if there was no wind?

$33$ - A Rowing Puzzle

A crew can row a certain course upstream in $8 \tfrac 4 7$ minutes,

and, if there were no stream, they could row it in $7$ minutes less than it takes them to drift down the stream.

How long would it take to row down with the stream?

$34$ - The Moving Stairway

On one of the moving stairways on the London Tube Railway I find that if I walk down $26$ steps I require $30$ seconds to get to the bottom,

but if I take $34$ steps I require only $18$ seconds to reach the bottom.

What is the height of the stairway in steps?

$35$ - Sharing a Bicycle

Anderson and Brown have to go $20$ miles and arrive at exactly the same time.

They have only one bicycle.

Anderson can only walk $4$ miles an hour,

while Brown can walk $5$ miles an hour,

but Anderson can ride $10$ miles an hour to Brown's $8$ miles an hour.

How are they to arrange the journey?

$36$ - More Bicycling

Referring to the last puzzle, let us now consider the case where a third rider has to share the same bicycle.

As a matter of fact, I understand that Anderson and Brown have taken a man named Carter into partnership, and the position today is this:

Anderson, Brown and Carter walk respectively $4$, $5$ and $3$ miles per hour,

and ride respectively $10$, $8$ and $12$ miles per hour.

How are they to use that single bicycle so that all shall complete the $20$ miles journey at the same time?

$37$ - A Side-car Problem

Atkins, Baldwin and Clarke had to go a journey of $52$ miles across country.

Atkins had a motor-bicycle with sidecar for one passenger.

How was he to take one of his companions a certain distance,

drop him on the road to walk the remainder of the way,

and return to pick up the second friend,

so that they should all arrive at their destination at exactly the same time?

The motor-bicycle could do $20$ miles per hour,

Baldwin could walk $5$ miles per hour,

and Clarke could walk $4$ miles per hour.

$38$ - The Despatch-Rider

If an army $40$ miles long advances $40$ miles

while a despatch-rider gallops from the rear to the front,

delivers a despatch to the commanding general,

and returns to the rear,

how far has he to travel?

$39$ - The Two Trains

Two railway trains, one $400$ feet long and the other $200$ feet long, ran on parallel rails.

It was found that when they went in opposite directions they passed each other in $5$ seconds,

but when they ran in the same direction the faster train would pass the other in $15$ seconds.

Now, a curious passenger worked out from these facts the rate per hour at which each train ran.

Can the reader discover the correct answer?

$40$ - Pickleminster to Quickville

Two trains, $A$ and $B$, leave Pickleminster for Quickville at the same time as two trains, $C$ and $D$, leave Quickville for Pickleminster.

$A$ passes $C$ $120$ miles from Pickleminster and $D$ $140$ miles from Pickleminster.

$B$ passes $C$ $126$ miles from Quickville and $D$ half-way between Pickleminster and Quickville.

Now, what is the distance from Pickleminster to Quickville?

$41$ - The Damaged Engine

We were going by train from Anglechester to Clinkerton, and an hour after starting some accident happened to the engine.

We had to continue the journey at $\tfrac 3 5$ of the former speed, and it made us $2$ hours late at Clinkerton,

and the driver said that if only the accident had happened $50$ miles farther on the train would have arrived $40$ minutes sooner.

Can you tell from that statement just how far it is from Anglechester to Clinkerton?

$42$ - The Puzzle of the Runners

Two men ran a race round a circular course, going in opposite directions.

Brown was the best runner and gave Tompkins a start of $\tfrac 1 8$ of the distance.

But Brown, with a contempt for his opponent, took things too easily at the beginning,

and when he had run $\tfrac 1 6$ of his distance he met Tompkins,

and saw that his chance of winning the race was very small.

How much faster than he went before must Brown now run in order to tie with his competitor?

$43$ - The Two Ships

A correspondent asks the following question.

Two ships sail from one port to another -- $200$ nautical miles -- and return.

The Mary Jane travels outwards at $12$ miles an hour and returns at $8$ miles an hour,

thus taking $41 \tfrac 2 3$ hours on the double journey.

The Elizabeth Ann travels both ways at $10$ miles an hour, taking $40$ hours on the double journey.

Now, seeing that both ships travel at the average speed of $10$ miles per hour, why does the Mary Jane take longer than the Elizabeth Ann?

$44$ - Find the Distance

A man named Jones set out to walk from $A$ to $B$,

and on the road he met his friend Kenward, $10$ miles from $A$, who had left $B$ at exactly the same time.

Jones executed his commission at $B$ and, without delay, set out on his return journey,

while Kenward as promptly returned from $A$ to $B$.

They met $12$ miles from $B$.

Of course, each walked at a uniform rate throughout.

Now, how far is $A$ from $B$?

$45$ - The Man and the Dog

"Yes, when I take my dog for a walk," said a mathematical friend, "he frequently supplies me with some interesting problem to solve.

One day, for example, he waited, as I left the door, to see which way I should go,

and when I started he raced to the end of the road, immediately returning to me;

again racing to the end of the road and again returning.

He did this four times in all, at a uniform speed,

then ran at my side the remaining distance, which according to my paces measured $27$ yards.

I afterwards measured the distance from my door to the end of the road and found it to be $625$ feet.

Now, if I walk $4$ miles per hour, what is the speed of my dog when racing to and fro?"

$46$ - Baxter's Dog

Anderson set off from an hotel at San Remo at nine o'clock and had been walking an hour when Baxter went after him along the same road.

Baxter's dog started at the same time as his master and ran uniformly forwards and backwards between him and Anderson until the two men were together.

Anderson's speed is $2$, Baxter's $4$, and the dog's $10$ miles an hour.

How far had the dog run when Baxter overtook Anderson?

$47$ - The Runner's Refreshment

A man runs $n$ times round a circular track whose radius is $t$ miles.

He drinks $s$ quarts of beer for every mile that he runs.

Prove that he will only need one quart!

$48$ - Railway Shunting

How are the trains in our illustration to pass one another, and proceed with their engines in front?

The small side track is large enough to hold one engine or one carriage at a time, and no tricks, such as ropes and flying-switches, are allowed.

Every reversal -- that is, change of direction -- of an engine is counted as a move in the solution.

What is the smallest number of moves necessary?

$49$ - Exploring the Desert

Nine travellers, each possessing a motor-car, meet on the eastern edge of a desert.

They wish to explore the interior, always going due west.

Each car can travel $40$ miles on the contents of the engine tank,

which holds a gallon of petrol,

and each can carry $9$ extra gallon tins of petrol and no more.

Unopened tins can alone be transferred from car to car.

What is the greatest distance at which they can enter the desert without making any depots of petrol for the return journey?

$50$ - Exploring Mount Neverest

Professor Walkingholme, one of the exploring party, was allotted the special task of making a complete circuit of the base of the mountain at a certain level.

The circuit was exactly $100$ miles in length and he had to do it all alone on foot.

He could walk $20$ miles a day, but he could only carry rations for $2$ days at a time,

the rations for each day being packed in sealed boxes for convenience in dumping.

He walked his full $20$ miles every day and consumed $1$ day's ration as he walked.

What is the shortest time in which he could complete the circuit?