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Teaching Mathematics Foundations to Middle Years. Woo's Wonderful World of Maths. The Art of Statistics Learning from Data. Antifragile Things That Gain from Disorder. The Maths of Life and Death. Item Added: Puzzles and Curious Problems. Howold is little Mike? THEIR AGES Rackbrane said the other morning that a man on being asked the ages ofhis two sons stated that eighteen more than the sum of their ages is doublethe age of the elder, and six less than the difference of their ages is the age ofthe younger.

What are their ages? Age Puzzles 13four times as old; last year I was three times as old; and this year I am twoand one-half times as old. What was the age ofeach? Every age was an exact number of years. The readermay think, at first sight, that there is insufficient data for an answer, but hewill be wrong: A man's age at death was one twenty-ninth of the year of his birth. Howold was he in the year ? This tempted us to work out the day of his birth. Perhapsthe reader may like to do the same. We will assume he was born at midday. Their united ages that is, the combined years of their complete lives were one hundred years.

Cleopatra died 30 B. When was Boadicea born? One little incident was fresh in my memory when I awakened.

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Isaw a clock and announced the time as it appeared to be indicated, butmy guide corrected me. Except for this improvement, ourclocks are precisely the same as. At what time are the two hands of a clock so situated that, reckoningas minute points past XII, one is exactly the square of the distance of the other? If it was set going at noon, what would be the first timethat it would be impossible, by reason of the similarity of the hands, to besure of the correct time?

Readers will remember that with these clock puzzles there is the conventionthat we may assume it possible to indicate fractions of seconds.


On thisassumption an exact answer can be given. George fell into the trap that catches so manypeople, of writing the fourth hour as IV, instead of I1II. Clock Puzzles 15 Colonel Crackham then asked themto show how a dial may be broken intofour parts so that the numerals oneach part shall in every case sum to As an example he gave our illustra-tion, where it will be found that theseparated numerals on two parts sumto 20, but on the other parts they addup to 19 and 21 respectively, so it fails.

But it was found that they hadreally only changed places. As you know, the dancing commenced betweenten and eleven oclock. What was the exact time of the start? What was the correct time? On his return between fourand five oclock he noticed that the hands were exactly reversed. What werethe exact times that he made the two crossings?

How far was it to the topof the hill? Ackworth, who is leading, went up three steps at a time, asarranged; Barnden, the second man, went four steps at a time, and Croft, who. Undoubtedly Ackworth wins. But the point is, howmany 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 wantedto show the finish.

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But it is possible to tell from the evidence the fewest pos-sible steps in that staircase. Can you do it? They met on the road at five minutes past four oclock, andeach man reached his destination at exactly the same time. Can you sayat what time they both arrived? How long would ittake him to ride a mile if there was no wind?

Some will say that the average. That answer is entirely wrong. How long would it take torow down with the stream? What isthe height of the stairway in steps? The time is measured from the momentthe top step begins to descend to the time I step off the last step at the bottomonto the level platform. They hadonly a single bicycle, which they rode in turns, each rider leaving it in thehedge when he dismounted for the one walking behind to pick up, and walk-ing ahead himself, to be again overtaken.

What was their best way of arrang-ing their distances? As their walking and riding speeds were the same, itis extremely easy. Simply divide the route into any even number of equal stagesand drop the bicycle at every stage, using the cyclometer. Each man wouldthen walk half way and ride half way. But here is a case that will require a little more thought. Anderson and Brownhave to go twenty miles and arrive at exactly the same time.

They have onlyone bicycle. Anderson can only walk four miles an hour, while Brown canwalk five miles an hour, but Anderson can ride ten miles an hour to Brown'seight miles an hour. How are they to arrange the journey? Each man always either walks orrides at the speeds mentioned, without any rests. As a matter of fact, I understand that Andersonand Brown have taken a man named Carter into partnership, and the positiontoday is this: Anderson, Brown, and Carter walk respectively four, five, andthree miles per hour, and ride respectively ten, eight, and twelve miles perhour.

How are they to use that single bicycle so that all shall completethe twenty miles' journey at the same time? Atkins had a motorcycle with a sidecar for one passenger. How was he to take one of his companions a certain distance, drop him on theroad to walk the remainder of the way, and return to pick up the second friend,who, starting at the same time, was already walking on the road, so that theyshould all arrive at their destination at exactly the same time? The motorcycle could do twenty miles an hour, Baldwin could walk fivemiles an hour, and Clarke could walk four miles an hour.

Of course, each wentat his proper speed throughout and there was no waiting. I might have complicated the problem by giving more passengers, butI have purposely made it easy, and all the distances are an exact number ofmiles-without fractions.

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It was found that when they went in oppositedirections they passed each other in five seconds, but when they ran in the samedirection the faster train would pass the other in fifteen seconds. A curious.

Can the reader discover the correct answer? Of course, each train ran witha uniform velocity. A passes C milesfrom Pickleminster and D miles from Pickleminster. Now,what is the distance from Pickleminster to Quickville? Every train runs uni-formly at an ordinary rate. We had to continue the journey at three-fifths of the former speed. It madeus two hours late at Clinkerton, and the driver said that if only the accidenthad happened fifty miles farther on the train would have arrived forty minutessooner.

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Can you tell from that statement just how far it is from Anglechesterto Clinkerton? Brown was the best runner and gave Tompkins a start of one-eighth of thedistance. But Brown, with a contempt for his opponent, took things too easilyat the beginning, and when he had run one-sixth of his distance he metTompkins, 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 tiewith his competitor? The puzzle is quite easy when once you have grasped itssimple conditions. Two ships sail from one portto another-two hundred nautical miles-and return. The Mary Jane travels. The ElizabethAnn travels both ways at ten miles an hour, taking forty hours on the doublejourney.

Seeing that both ships travel at the average speed often miles per hour, whydoes the Mary Jane take longer than the Elizabeth Ann? Perhaps the readercould explain this little paradox. Jones executed his commissionat B and, without delay, set out on his return journey, while Kenwardas promptly returned from A to B. They met twelve milesfrom B. Of course, each walked at a uniform rate throughout. Howfar is A from B? I will show the reader a simple rule by which the distance may be foundby anyone in a few seconds without the use of a pencil. In fact, it is quiteabsurdly easy-when you know how to do it.

One day, forexample, he waited, as I left the door, to see which way I should go, and whenI started he raced along 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 timesin all, at a uniform speed, and then ran at my side the remaining distance,which according to my paces measured 27 yards. I afterwards measured thedistance from my door to the end of the road and found it to be feet.

Now,ifl walk 4 miles per hour, what is the speed ofmy dog when racing to and fro? Andersonset off from an hotel at San Remo at nine oc1ock and had been walking an. Baxter's dog startedat the same time as his master and ran uniformly forwards and backwardsbetween him and Anderson until the two men were together.

Anderson'sspeed is two, Baxter's four, and the dog's ten miles an hour. How far had thedog run when Baxter overtook Anderson? He drinkss quarts of beer for every mile that he runs. Prove that he will only need onequart! They wish to explore the interior, always going due west. Each car can travelforty 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. Unopenedcans can alone be transferred from car to car. What is the greatest distanceat which they can enter the desert without making any depots of fuel for thereturn journey?

The circuit was exactly a hundred miles in length and he had to do itall alone on foot.

536 puzzles & curious problems.

He could walk twenty miles a day, but he could only carryrations for two days at a time, the rations for each day being packed in sealedboxes for convenience in dumping. He walked his full twenty miles every dayand consumed one day's ration as he walked. What is the shortest timein which he could complete the circuit? This simple question will be found to form one of the most fascinating. It made a considerable demandon Professor Walkingholme's well-known ingenuity. The idea was suggestedto me by Mr. He is an invalid, and at12 noon started in his Bath chair from B towards C.

His friend, who hadarranged to join him and help push back, left A at 15 P. He joined him, and with his help they went backat four miles per hour, and arrived at A at exactly 1 P. How far did ourcorrespondent go towards C? A passenger at theback of the train wishes to walk to the front along the corridor and in doingso walks at the rate of three miles per hour. At what rate is the man travellingover the permanent way? We will not involve ourselves here in quibbles anddifficulties similar to Zeno's paradox of the arrow and Einstein's theory ofrelativity, but deal with the matter in the simple sense of motion in referenceto the permanent way.

Right away! What is the correct number of trains? His gardener and a boy both insisted on carrying the luggage; butthe gardener is an old man and the boy not sufficiently strong, while thegentleman 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 shouldshare the burden equally among them.

How would you have managed it? It isassumed that each man travelled at a uniform rate, and the speed of thestaircase was also constant. The four cyclists started at noon.

536 Puzzles and Curious Problems

Each person rode round a different circle, one at the rate of six miles an hour,another at the rate of nine miles an hour, another at the rate of twelve milesan hour, and the fourth at the rate of fifteen miles an hour. They agreed to ride. The distance round each circle was exactly one-third of a mile.

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When did theyfinish their ride? Atkins could walk one mile an hour, Brown could walktwo miles an hour, and Cranby could go in his donkey cart at eight miles anhour. Cranby drove Atkins a certain distance, and, dropping him to walk theremainder, drove back to meet Brown on the way and carried him to theirdestination, where they all arrived at the same time.

Of course each went at a uniform ratethroughout. Andrews is acertain distance behind Brooks, and Carter is twice that distance in front ofBrooks. Each car travels at its own uniform rate of speed, with the result thatAndrews passes Brooks in seven minutes, and passes Carter five minutes later. In how many minutes after Andrews would Brooks pass Carter?

A car, A, starts at noon from one end and goesthroughout at 50 miles an hour, and at the same time another car, B, goinguniformly at miles an hour, starts from the other end together with a flytravelling miles an hour. When the fly meets car A, it immediately turnsand flies towards B. The fly then turns towards A and continues flying backwards and forwardsbetween A and B. A bit of exercise, you know. But this is the long-est stairway on the line-nearly a thousand steps. I will tell you a queer thingabout it that only applies to one other smaller stairway on the line.

If I go uptwo steps at a time, there is one step left for the last bound; if I go up threeat a time, there are two steps left; ifl go up four at a time, there are three stepsleft; five at a time, four are left; six at a time, five are left; and if I went upseven at a time there would be six steps left over for the last bound. Now,why is that? The bottom floor does not count as a step, and thetop landing does. Now, if the bus goes atthe rate of nine miles an hour and they walk at the rate of three miles anhour, how far can they ride so that they may be back in eight hours?

They requi-sitioned the services of a man with a small car. At what rate can you walk? So all you have to do is to keep walking while you are on your feet, andI will do the res t. What was the exact time that they all arrived together? So, of course, she took five hours longer in coming up thanin going down. What was it? Wilkinson, walks from hiscountry 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 threemiles per hour.

The double journey takes him exactly seven hours. Can youtell the distance from his house to the town? This friend was to leave home at thesame time and ride to London to put up at the Crackhams' house. They tookthe same route, and each car went at its own uniform speed. They kept alook-out for one another, and met forty miles from Bugleminster.

Brown can ride once round the trackin six minutes, and Robinson in four minutes. In how many minutes willRobinson overtake Brown? How far apart are theyexactly an hour before they meet? Because I have failed to find these cities on any map or in any gazetteer, Icannot state the distance between them, so we will just assume that it is morethan miles. But the only convey-ance obtainable was old Pat Doyle's rickety little cart, propelled by a marewhose working days, like her legs, were a bit over.

She wouldn't slow down and she wouldn't put on any spurts.

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I got exactly the same reply. It was clear he could only think in terms of Pigtown. They each walked yards in adirect line, with their faces towards each other, and you would suppose thatthey must have met. Yet they found after their yards' walk that they werestill yards apart. Can you explain?

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What was its true weight? It will be seen from the illustration below that one arm is longer than theother, though they are purposely so drawn as to give no clue to the answer. As a consequence, it happened that in one of the cases exhibited eight of thelittle packets it does not matter what they contain exactly balanced three ofthe canisters, while in the other case one packet appeared to be of the sameweight as six canisters.

The true weight of one canister was known to be exactly one ounce. Whatwas the true weight of the eight packets? Weight Puzzles 31task of weighing the baby. Whenever We must try to work it out at home. What do yourolled off, while the father was holding suppose was the actual weight of thatoff the dog, who always insisted on dear infant?

Atlast the man with the baby and Fidowere on the machine together, and Itook this snapshot of them with mycamera. Can you say how many plums were equal in weight to one pear? The relative sizes of the fruits in the drawing must not be taken to be correct they are purposely not so , but we must assume that every fruit is exactlyequal in weight to every other of its own kind.

It is clear that three apples and one pear are equal in weight to ten plums,and that one apple and six plums weigh the same as a single pear, but howmany plums alone would balance that pear? When the novice starts working it out he willinevitably be adopting algebraical methods, without, perhaps, being consciousof the fact. The two weighings show nothing more than two simultaneousequations, with three unknowns. What is the quickest way for him to do the business? We will say at once that only nine weighings are really necessary. Thus, if it were1 2 8 9 6, then 12 multiplied by 8 produces But, unfortunately, I, 2, 6, 8, 9are not successive numbers, so it will not do.

How can I arrange right if that 2 had happened to bethem in a row so that the number another 4. Of course, there must beformed by the first pair multiplied by two solutions, for the pairs are clearlythe number formed by the last pair, interchangeable. Thus, in theexample I have shown, 31 multipliedby 79 and 5 subtracted will produce Thus the greatest possible number might be. Digital Puzzles 33five and the smallest square number with five similar digits at the end mightbe But this is certainly not a square number.

Of course, 0 is not tobe regarded as a digit. For example, 1,2,7,8 and 3, 4,5,6 add up to 18 in bothcases. But you are not allowed to make any such reversal. What is the largest number that we can multiply by in order to produce a similar product of eight figures with the first fourfigures repeated in the same order?

There is no objection to a repetition offigures-that is, the four that are repeated need not be all different, as in thecase shown. Can you find a number begin-ning with 7 that can be divided by 7 in the same simple manner? He has apparently come across our last puzzle withthe conditions wrongly stated.

If you are to transfer the first figure to the endit is solved by 3 I 5 7 8 9 4 7 3 6 84 2 I 0 5 2 6, and a solution may easily befound from this with any given figure at the beginning. But if the figure is tobe moved from the end to the beginning, there is no possible solution for thedivisor 2. But there is a solution for the divisor 3. Can you find it? It has since been dealt with at some length byvarious writers. The point is to express all possible whole numbers with fourfours no more and no fewer , using the various arithmetical signs.

All numbers up to inclusive may be solved, using only the signs for addition, subtraction, multi-plication, division, square root, decimal points, and the factorial sign 4! It is necessary to discover which numbers can be formed with one four,with two fours, and with three fours, and to record these for combination asrequired. It is the failure to find some of these that leads to so much difficulty. For example, I think very few discover that 64 can be expressed with onlytwo fours. Can the reader do it? If I multiply and add 2 and 47, I get 94 and the same figures.

IfI multiply and add 3 and 24, I get the same figures and Can you findtwo numbers that when multiplied and added will, in this simple manner,produce the same three figures? There are two cases. Some of them are. Digital Puzzles 35very easily found.