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Speed and Distance Puzzles on Mental Calculation

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

Correct! Wrong!

The distance must have been 300 miles.

If an army forty miles long advances forty miles while a dispatch rider gallops from the rear to the front, delivers a dispatch to the commanding general, and returns to the rear, how far has he to travel?

Correct! Wrong!

The answer is the square root of twice the square of 40, added to 40. This is 96.568 miles, or, roughly, 96~ miles.

On one of the escalators on the London subway I find that if I walk down twenty-six steps I require thirty seconds to get to the bottom, but if I make thirty-four steps I require only eighteen seconds to reach the bottom. What is the height of the stairway in steps? The time is measured from the moment the top step begins to descend to the time I step off the last step at the bottom onto the level platform.

Correct! Wrong!

If I walk 26 steps I require 30 seconds, and if I walk 34 steps I require only 18 seconds. Multiply 30 by 34 and 26 by 18 and we get 1,020 and 468, the difference between which is 552. Divide this by the difference between 30 and 18 (that is, by 12) and the answer is 46, the number of steps in the stairway, which descends at the rate of 1 step in 1 ~ seconds. The speed at which I walk on the stairs does not affect the question, as the step from which I alight will reach the bottom at a given moment, whatever I do in the meantime.

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!

Correct! Wrong!

As the radius is t, the diameter is 2t. The diameter multiplied by 'TT (the Greek letter pi) gives us the circumference, 2t'TT miles. As he goes round n times, 2t'TTn equals the number of miles run, and, as he drinks s quarts per mile, he consumes 2t'TTns quarts. We can put the factors in any order: therefore the answer is 2'TTnts (two pints) or one quart.

George treated his best girl to a bus ride, 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 that they may be back in eight hours?

Correct! Wrong!

They can ride three times as fast as they can walk, therefore three-quarters of their time must have been spent in walking, and only a quarter in riding. Therefore they rode for 2 hours, going 18 miles, and walked back in 6 hours, thus exactly filling their 8 hors.

Nine travellers, each possessing a car, meet on the eastern edge of a desert. They wish to explore the interior, always going due west. Each car can travel forty miles on the contents of the engine tank, which holds a gallon of fuel, and each can carry nine extra gallon cans of fuel and no more. Unopened cans 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 fuel for the return journey?

Correct! Wrong!

The nine men, A, B, C, D, E, F, G, H, J, all go 40 miles together on the I gal. in their engine tanks, when A transfers I gal. to each of the other eight and has I gal. left to return home. The eight go another 40 miles, when B transfers I gal. to each of the other seven and has 2 gals. to take him home. The seven go another 40 miles, when C transfers I gal. to each ofthe six others and returns home on the remaining 3 gals. The six go another 40 miles, when D gives each of five I gal. and returns home. The five go 40 miles, when E gives each of four I gal. and returns home. The four go another 40 miles, when F gives each of three I gal. and returns home. The three go 40 miles, when G gives each of two I gal. and returns home. The two go 40 miles, when H gives I gal. to J and returns home. Finally, the last man, J, goes another 40 miles and then has 9 gals. to take him home. Thus J has gone 360 miles out and home, the greatest distance in a straight line that could be reached under the conditions.

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 after their 200 yards' walk that they were still 400 yards apart. Can you explain?

Correct! Wrong!

The second man, on seeing his friend turn and walk towards him, walked backwards 200 yards. It was an eccentric thing to do, but he did it, and it is the only answer to the puzzle. They could thus have their faces towards each other and be going in a direct line.

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 four, five, and three miles per hour, and ride respectively ten, eight, and twelve miles per hour. How are they to use that single bicycle so that all shall complete the twenty miles' journey at the same time?

Correct! Wrong!

A. rides 71 Y:17 miles, B. rides l 13b miles, and C. rides I Ph7 miles, making the 20 miles in all. They may ride in any order, only each man should complete his ride in one mount and the second rider must always walk both before and after riding. They will each take 3% hours on the journey, and therefore will all arrive together.

A gentleman had to walk to his railway station, four miles from his house, and was encumbered by two bags of equal weight, but too heavy for him to carry alone. His gardener and a 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 oflabor 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. How would you have managed it?

Correct! Wrong!

Let the boy continue to carry one bag for I \-) miles; then hand it to the gentleman, who will carry it to the station. Also let the man carry his bag 27'3 miles and then deliver it to the boy, who will carry it for the remaining distance. Then each of the three persons will have carried one bag 27'3 miles -an equal division of labor.

Three 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. In how many minutes after Andrews would Brooks pass Carter?

Correct! Wrong!

B would pass C in 613 minutes.

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?

Correct! Wrong!

They will come together 12 minutes from the start.

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 at forty-eight miles from London." If this were so, what is the distance from London to Bugleminster?

Correct! Wrong!

The distance from London to Bugleminster must be 72 miles.

Weary Willie went up a certain hill at the rate of one and a half miles per hour and came down at the rate of four and a half miles per hour, so that it took him just six hours to make the double journey. How far was it to the top of the hill?

Correct! Wrong!

.It must have been 6 3/4 miles to the top of the hill. He would go up in 4~ hours and descend in I ~ hours

A train is travelling at the rate of sixty 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? We will not involve ourselves here in quibbles and difficulties similar to Zeno's paradox of the arrow and Einstein's theory of relativity, but deal with the matter in the simple sense of motion in reference to the permanent way.

Correct! Wrong!

Assume that the train runs for an hour, and that it is itself of the absurd length of 3 miles. Then, as in the diagram, the train will have gone from B to C = 60 miles, but the passenger will have gone from A to C, or 63 miles. On the other hand, if he walks from the front to the rear of the train, the train will have gone from B to C (again 60 miles), while the passenger will have gone from B to D = 57 miles. So that in our first case the man would be travelling over the permanent way at the rate of 63 miles per hour, and in the second case, 57 miles per hour.

We were going by train from Anglechester to Clinkerton, and an hour after starting an accident happened to the engine. We had to continue the journey at three-fifths of the former speed. It made us two hours late at Clinkerton, and the driver said that if only the accident had happened fifty miles farther on the train would have arrived forty minutes sooner. Can you tell from that statement just how far it is from Anglechester to Clinkerton?

Correct! Wrong!

The distance from Angiechester to Clinkerton must be 200 miles. The train went 50 miles at 50 m.p.h. and 150 miles at 30 m.p.h. If the accident had occurred 50 miles farther on, it would have gone 100 miles at 50 m.p.h. and 100 miles at 30 m.p.h

Colonel Crackham says that his friend, Mr. Wilkinson, walks from his country house into the neighboring town at the rate of five miles per hour, and, because he is a little tired, he makes the return journey at the rate of three miles per hour. The double journey takes him exactly seven hours. Can you tell the distance from his house to the town?

Correct! Wrong!

The distance must be 13 Ii miles, so that he walked into the town in 2% hours and returned in 4% hours, making 7 hours, as stated.

A crew can row a certain course upstream in eight and four-sevenths minutes, and, if there were no stream, they could row it in seven minutes less than it takes them to drift down the stream. How long would it take to row down with the stream?

Correct! Wrong!

The correct answer is 3 9/17 minutes. The crew can row Vs of the distance per minute on still water, and the stream does y\2 of the distance per minute. The difference and sum of these two fractions are ~o and 1 ~o. Therefore, against the stream would take 61)7 minutes (or 817 minutes), and with the stream 6iJ'b (or 3o/i7 minutes).

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