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

Henry Ernest Dudeney: Puzzles and Curious Problems: Arithmetical and Algebraical Problems

$64$ - The Bath Chair

A correspondent informs us that a friend's house at $A$, where he was invited to lunch at $1$ p.m., is a mile from his own house at $B$.

He is an invalid, and at $12$ noon started in his Bath chair from $B$ towards $C$.

His friend, who had arranged to join him and help push back, left $A$ at $12.15$ p.m., walking at $5$ miles per hour towards $C$.

He joined him, and with his help they went back at $4$ miles per hour, and arrived at $A$ at exactly $1$ p.m.

How far did our correspondent go towards $C$?

$65$ - The Pedestrian Passenger

A train is travelling at the rate of $60$ miles per hour.

A passenger at the back of the train wishes to walk to the front along the corridor,

and in doing so walks at the rate of three miles per hour.

At what rate is the man travelling over the permanent way?

$66$ - Meeting Trains

At Wurzeltown Junction an old lady put her her head out of the window and shouted:

"Guard! how long will the journey be from here to Mudville?"

"All the trains take five hours ma'am, either way," replied the official.

"And how many trains shall I meet on the way?"

This absurd question tickled the guard, but he was ready with his reply:

"A train leaves Wurzletown for Mudville, and also one from Mudville to Wurzletown, at five minutes past every hour. Right away!"

The old lady induced one of her fellow passengers to work out the answer for her.

What is the correct number of trains?

$67$ - Carrying Bags

A gentleman had to walk to his railway station, four miles from his house,

and was encumbered by two heavy bags of equal weight, but too heavy for him to carry alone.

His gardener and the boy both insisted on carrying the luggage;

but the gardener is an old man, and the boy not sufficiently strong,

while the gentleman believes in a fair division of labour, and wished to take his own share.

They started off with the gardener carrying one bag and the boy the other,

while the gentleman worked out the best way of arranging that the three should share the burden equally among them.

Now, how would you have managed it?

$68$ - The Moving Staircase

"I counted $50$ steps that I made in going down the moving staircase," said Walker.

"I counted $75$ steps," said Trotman; "but I was walking down three times as quickly as you."

If the staircase were stopped, how many steps would be visible?

$69$ - The Four Cyclists

The four circles represent cinder paths.

The four cyclists started at noon.

Each person rode round a different circle,

one at the rate of $6$ miles an hour,

another at the rate of $9$ miles an hour,

another at the rate of $12$ miles an hour,

and the fourth at the rate of $15$ miles an hour.

They agreed to ride until all met at the centre, from which they started, for the fourth time.

The distances around each circle was exactly one-third of a mile.

When did they finish their ride?

$70$ - The Donkey Cart

Atkins, Brown and Cranby had to go an journey of $40$ miles.

Atkins could walk $1$ mile an hour,

Brown could walk $2$ miles an hour,

and Cranby could go in his donkey-cart at $8$ miles an hour.

Cranby drove Atkins a certain distance, and, dropping him to walk the remainder,

drove back to meet Brown on the way and carried him to their destination,

where they all arrived at the same time.

How long did the journey take?

$71$ - The Three Motor-Cars

Three motor-cars travelling along a road in the same direction are, at a certain moment, in the following positions in relation to one another.

Andrews is a certain distance behind Brooks,

and Carter is twice that distance in front of Brooks.

Each car travels at its own uniform rate of speed,

with the result that Andrews passes Brooks in seven minutes,

and passes Carter five minutes later.

Now, in how many minutes after Andrews would Brooks pass Carter?

$72$ - The Fly and the Motor-Cars

A road is $300$ miles long.

A motor-car, $A$, starts at noon from one end and goes throughout at $50$ miles an hour,

and at the same time another car, $B$, going uniformly at $100$ miles an hour, starts from the other end,

together with a fly travelling $150$ miles an hour.

When the fly meets the car $A$, it immediately turns and flies towards $B$.

$(1)$ When does the fly meet $B$?

The fly then turns towards $A$ and continues flying backwards and forwards between $A$ and $B$.

$(2)$ When will the fly be crushed between the two cars if they collide and it does not get out of the way?

$73$ - The Tube Stairs

We ran up against Percy Longman, a young athlete, the other day when leaving Curley Street tube station.

He stopped at the lift, saying, "I always go up by the stairs.

A bit of exercise, you know.

But this is the longest stairway on the line -- nearly $1000$ steps.

I will tell you a queer thing about it that only applies to one other smaller stairway on the line.

If I go up two steps at a time, there is one step left for the last bound;

if I go up three at a time, there are two steps left;

if I go up four at a time, there are three steps left;

five at a time, four are left;

six at a time, five are left;

and if I went up seven at a time there would be six risers left over for the last bound.

Now, why is that?"

As he went flying up the stairs, three steps at a time, we laughed and said,

He little suspects that if he went up twenty steps at a time there would be nineteen risers for his last bound!"

How many risers are there in the Curley Street tube stairway?

The platform does not count as a riser, and the top landing does.

$74$ - The Omnibus Ride

George treated his best girl to a ride on a motor omnibus,

but on account of his limited resources it was necessary that they should walk back.

Now, if the bus goes at the rate of nine miles an hour and they walk at the rate of three miles an hour,

how far can they ride so they may be back in eight hours?

$75$ - A Question of Transport

Twelve soldiers had to get to a place twenty miles distant with the quickest possible dispatch,

and all had to arrive at the same time.

They requisitioned the services of a man with a small motor-car.

"I can do twenty miles an hour," he said, "but I cannot carry more than four men at a time.

At what rate do you walk?"

"All of us can do a steady four miles an hour," they replied.

"Very well," exclaimed the driver, "then I will go ahead with four men,

drop them somewhere on the road to walk,

then return and pick up four more (who will then be somewhere on the road),

drop them off also, and return for the last four.

So all you have to do is to keep walking while you are on your feet, and I will do the rest."

As they started at noon, what was the exact time that they all arrived together?

$76$ - How Far Was It?

"The steamer," remarked one of our officers home from the East, "was able to go twenty miles an hour down-stream,

but could only do fifteen miles an hour upstream.

So, of course, she took five hours longer in coming up than in going down."

One could not resist working out mentally the distance from point to point.

What was it?

$77$ - Out and Home

Mr Wilkinson walks from his country house into the neighbouring town at the rate of five miles per hour,

and, as he is a little tired, he makes the return journey at the rate of three miles per hour.

As the double journey takes him exactly seven hours, can you tell me the distance from his house to the town?

$78$ - The Meeting Cars

The Crackhams made their first stop at Bugleminster, where they were to spend the night at a friend's house.

This friend was to leave home at the same time and ride to London to put up at the Crackhams' house.

They took the same route, and each car went at its own uniform speed.

They kept a look-out for one another, and met forty miles from Bugleminster.

George that evening worked out the following little puzzle:

"I find that if, on our respective arrivals, we had each at once proceeded on the return journey at the same speeds

we should meet $48$ miles from London."

If this were so, what is the distance from London to Bugleminster?

$79$ - A Cycle Race

Two cyclists race on a circular track.

Brown can ride once round the track in six minutes,

and Robinson in four minutes.

In how many minutes will Robinson overtake Brown?

$80$ - A Little Train Puzzle

A non-stop express going sixty miles an hour starts from Bustletown for Ironchester,

and another non-stop express going forty miles an hour starts at the same time from Ironchester for Bustletown.

How far apart are they exactly an hour before they meet?

As I have failed to find these cities on any map or in any gazetteer, I cannot state the distance between them,

so we will just assume that it is somewhere over $250$ miles.

$81$ - An Irish Jaunt

Colonel Crackham was going from Boghooley to Ballyfoyne, using Pat Doyle's horse and cart,

which moved at a steady rate, but more slowly than would normally be expected.

After they had been on the road for $20$ minutes, they had travelled half as far from Boghooley than it was to Pigtown.

They stopped for refreshment at Pigtown when they arrived there.

Five miles further on, it was half as far to Ballyfoyne as it was from Pigtown.

After another hour they had arrived in Ballyfoyne.

What is the distance from Boghooley to Ballyfoyne?

$82$ - A Walking Problem

A man taking a walk in the country on turning round saw a friend of his walking $400$ yards behind in his direction.

They each walked $200$ yards in a direct line, with their faces towards each other,

and you would suppose that they must have met.

Yet they found that after their $200$ yards walk that they were still $400$ yards apart.

Can you explain?