This is a classic type of puzzle from Henry Dudeney.
“This is a rough sketch of the finish of a race up a staircase in which three men took part. Ackworth, who is leading, went up three steps at a time, as arranged; Barnden, the second man, went four steps at a time, and Croft, who is last, went five at a time. Undoubtedly Ackworth wins. But the point is, how many steps are there in the stairs, counting the top landing as a step?
I have only shown the top of the stairs. There may be scores, or hundreds, of steps below the line. It was not necessary to draw them, as I only wanted to show the finish. But it is possible to tell from the evidence the fewest possible steps in that staircase. Can you do it?”
See the Staircase Race
This is another simple problem from H. E. Dudeney.
“103. PAINTING THE LAMP-POSTS.
Tim Murphy and Pat Donovan were engaged by the local authorities to paint the lamp-posts in a certain street. Tim, who was an early riser, arrived first on the job, and had painted three on the south side when Pat turned up and pointed out that Tim’s contract was for the north side. So Tim started afresh on the north side and Pat continued on the south. When Pat had finished his side he went across the street and painted six posts for Tim, and then the job was finished. As there was an equal number of lamp-posts on each side of the street, the simple question is: Which man painted the more lamp-posts, and just how many more?”
See Painting Lampposts
This typical problem from the prolific H. E. Dudeney may be a bit tricky at first.
“104.—CATCHING THE THIEF.
“Now, constable,” said the defendant’s counsel in cross-examination,” you say that the prisoner was exactly twenty-seven steps ahead of you when you started to run after him?”
“And you swear that he takes eight steps to your five?”
“That is so.”
“Then I ask you, constable, as an intelligent man, to explain how you ever caught him, if that is the case?”
“Well, you see, I have got a longer stride. In fact, two of my steps are equal in length to five of the prisoner’s. If you work it out, you will find that the number of steps I required would bring me exactly to the spot where I captured him.”
Here the foreman of the jury asked for a few minutes to figure out the number of steps the constable must have taken. Can you also say how many steps the officer needed to catch the thief?”
See Catching the Thief
This is another train puzzle by H. E. Dudeney. This one has some hairy arithmetic.
“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? Every train runs uniformly at an ordinary rate.”
See Trains – Pickleminster to Quickville
This is another delightful H. E. Dudeney puzzle.
“Mr. Gubbins, a diligent man of business, was much inconvenienced by a London fog. The electric light happened to be out of order and he had to manage as best he could with two candles. His clerk assured him that though both were of the same length one candle would burn for four hours and the other for five hours. After he had been working some time he put the candles out as the fog had lifted, and he then noticed that what remained of one candle was exactly four times the length of what was left of the other.
When he got home that night Mr. Gubbins, who liked a good puzzle, said to himself, ‘Of course it is possible to work out just how long those two candles were burning to-day. I’ll have a shot at it.’ But he soon found himself in a worse fog than the atmospheric one. Could you have assisted him in his dilemma? How long were the candles burning?”
See Mr. Gubbins in a Fog
Yet another train problem from H. E. Dudeney.
“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?”
See the Damaged Engine.
This is another train puzzle from H. E. Dudeney, which is fairly straight-forward.
“I put this little question to a stationmaster, and his correct answer was so prompt that I am convinced there is no necessity to seek talented railway officials in America or elsewhere. Two trains start at the same time, one from London to Liverpool, the other from Liverpool to London. If they arrive at their destinations one hour and four hours respectively after passing one another, how much faster is one train running than the other?”
See Two Trains – London to Liverpool
This is one of H. E. Dudeney’s train puzzles.
“Two railway trains, one four hundred feet long and the other two hundred feet long, ran on parallel rails. It was found that when they went in opposite directions they passed each other in five seconds, but when they ran in the same direction the faster train would pass the other in fifteen seconds. A curious passenger worked out from these facts the rate per hour at which each train ran. Can the reader discover the correct answer? Of course, each train ran with a uniform velocity.”
See Two Trains – Passing in the Night.
The following is a famous problem of Bachet as recounted by Heinrich Dörrie in his book 100 Great Problems of Elementary Mathematics:
“A merchant had a forty-pound measuring weight that broke into four pieces as the result of a fall. When the pieces were subsequently weighed, it was found that the weight of each piece was a whole number of pounds and that the four pieces could be used [in a balance scale] to weigh every integral weight between 1 and 40 pounds [when we are allowed to put a weight in either of the two pans]. What were the weights of the pieces?
(This problem stems from the French mathematician Claude Gaspard Bachet de Méziriac (1581-1638), who solved it in his famous book Problèmes plaisants et délectables qui se font par les nombres, published in 1624.)”
The problem has a nice solution using ternary numbers. See the Weight Problem of Bachet.
(Update 4/10/2019) Continue reading
This is a great puzzle by H. E. Dudeney involving a very useful technique.
“A man bought an odd lot of wine in barrels and one barrel containing beer. These are shown in the illustration, marked with the number of gallons that each barrel contained. He sold a quantity of the wine to one man and twice the quantity to another, but kept the beer to himself. The puzzle is to point out which barrel contains beer. Can you say which one it is? Of course, the man sold the barrels just as he bought them, without manipulating in any way the contents.”
See the Barrel of Beer.