CURIOUS CALCULATIONS

Rectangles

Solution

Take a piece of cardboard in the form of a Greek cross with arms, as shown here, and divide it by two straight cuts, so that the pieces when reunited form a perfect square.

Cross

Solution

The diagrams which we give below show how a hollow square can be formed of the pieces of three-quarters of another square from which a corner has been cutaway:—

Squares

The subjoined diagram shows how a square of paper or cardboard may be cut into nine pieces which, when suitably arranged, form five perfect squares.

Squares

On how many days can fifteen schoolgirls go out for a walk so arranged in rows of three, that no two are together more than once?

Fifteen schoolgirls can go out for a walk on seven days so arranged in rows of three that no two are together more than once.

It is said, on high authority, that there are no less than 15,567,522,000 different solutions to this problem. Here is one of them, given inBall’s Mathematical Recreations, in whichkstands for one of the girls, anda,b,c,d,e,f,g, in their modifications, for her companions on the seven differentdays:—

It is an excellent game of patience, for those who have time and inclination, to place the figures 1 to 15 inclusive in seven such columns, so as to fulfil the conditions.

It is possible from a Greek cross to cut off four equal pieces which, when put together, will form another Greek cross exactly half the size of the original, and by this process to leave a third Greek cross complete.

Crosses

This is how to doit:—

BisectC DatN,F GatO,K LatP, andB IatQ.

JoinN H,O M,P A,Q E, intersecting atR,S,T,U.

BisectA RatV,E SatW,T HatX, andM UatY.

JoinV Q,N Y,W P,O X,N W,V O,Q X,Y P.

Carefully cut out from the original Greek cross the newly-formed Greek cross, and the odd pieces from around it can be arranged to form another Greek cross.

The greatest number of plane figures that can be formed by the union of ten straight lines is thirty-six.

Crosses

The two equal lines at right angles are first drawn, and each is divided into eight equal parts. The other eight straight lines are then drawn fromatoa, frombtob, and so on, until the hammock-shaped network of thirty-six plane figures is produced.

It will be seen on the diagram below that seven straight vertical cuts with a table-knife will divide a crumpet into twenty-eight parts.

Crosses

The nine digits can be so adjusted as to form an equation, or, if taken as weight, to balance the scales.Thus:—

9, 61⁄2= 3, 5, 74⁄8

How large is the sea? This is a bold big question, and any possible answer involves a considerable stretch of the imagination. Here is a startling illustration of its vastvolume:—

If the water of the sea could be gathered into a round column, reaching the 93,000,000 miles which separate us from the sun, the diameter of this column would be nearly two miles and a half!

It is perhaps even more difficult to realise that this mighty mass of waters could be dissipated in a few moments, if the column we have imagined could become ice, and if the entire heat of the sun could be concentrated upon it. All would be melted inone second of time, and converted into steam in eight seconds!

Many of our readers may be glad to know an easy way to make an envelope of any shape or size.

Envelope

This diagram speaks for itself. When the linesA B,A C,D B,D Chave been drawn, the corners of the rectangle,H E G F, are folded over, as shown by the dotted lines, after the corners have been rounded, and the margins touched with gum.

The diagrams below will show how a piece of paper, 15 inches long and 3 inches wide, can be cut into five parts, and rearranged to form a perfect square.

Envelope

The following astounding calculation is answer enough to a question put by one of the authors of “Rejected Addresses:”—“Who filled the butchers’ shops with big blue flies?”

A pair of blow-flies can produce ten thousand eggs, which mature in a fortnight. If every egg hatches out, and there are equal numbers of either sex, which forthwith increase and multiply at the same rapid rate, and if their descendants do the like, so that all survive at the end of six months, it has been calculated that, if thirty-two would fill a cubic inch of space, the whole innumerable swarm would cover the globe, land and sea, half a mile deep everywhere.

Here is quite a neat way to make an equilateral triangle without usingcompasses:—

Triangle

Take a piece of paper exactly square, which we will callA B C D, fold it across the middle, so as to form the creaseE F; unfold it, and fold it again so that the cornerDfalls upon the creaseE FatG, and the angle atGis exactly divided. Again unfold the square, and fromGdraw the straight linesG CandG D. ThenG C Dis the equilateral triangle required.

How can we be sure that the value of a Frenchwoman is just 1 franc 8 centimes?

We can be sure that the exact value of a Frenchwoman is 1 franc 8 centimes, for

If a 52-feet ladder is set up so as just to clear a garden wall 12 feet high and 15 feet from the building, it will touch the house 48 feet from the ground.

Ladder

Our diagram shows this, and also, by a dotted line, the only other possible position in which it could fulfil the conditions, if it were then of any practical use.

If one pin could be dropped into a vessel this week, two the next, four the next, and so on, doubling each time for a year, the accumulated quantity would be 4,503,599,627,370,495, and their weight, if we reckon 200 pins to the ounce, would amount to 628,292,358 tons, a full load for 27,924 ships as large as theGreat Eastern, whose capacity was 22,500 tons.

When at the signpost which said “ToA4 miles, toB9 miles” on one arm, and on the other “ToC3 miles, toD—— miles,” and the boy whom I met could only tell me that the farm he worked at was equidistant fromA,B,C, andD, and nearer to them than to the signpost, and that all the roads ran straight, I found, thanks to memories of Euclid, that I was 12 miles fromD.

Map

SinceB AandD Cintersect outside the circle at the signpostE,

Q.E.D.

This seems to be quite a poor attempt at a Maltese cross, but there is method in the madness of its make.

Crooked cross

It is possible by two straight cuts to divide this uneven cross into four pieces which can be arranged together again so that they form a perfect square. Where must the cuts be made, and how are the four pieces rearranged?

Solution

We all remember that splendidly terse message of success sent home to the authorities by Napier when he had conquered the armies of Scinde—“Peccavi!” (I have sinned).

History had an excellent opportunity for repeating itself when Admiral Dewey defeated the Spaniel fleet, for he might have conveyed the news of his victory by the one burning word—“Cantharides”—“The Spanish fly!”

In the diagram below a square is subdivided into twenty-five cells.

Cells

Can you, keeping always on the straight lines, cut this into four pieces, and arrange these as two perfect squares, in which every semicircle still occupies the upper half of its cell?

Solution

The following very simple recipe for a home-made microscope has been suggested by a Fellow of the Royal MicroscopicalSociety:—

Take a piece of black card, make a small pinhole in it, put it close to the eye, and look at some small object closely, such as the type of a newspaper. A very decided magnifying power will be shown thus.

Where on this diagram must we place twenty-one pins or dots so that they fall into symmetrical design and form thirty rows, with three in each row?

Solution

Those who are fond of figures will find it a most interesting exercise to see how far they are able to represent every number, from one up to a hundred, by the use of four fours. Any of the usual signs and symbols of arithmetic may be brought into use. Here are a few instances of what may thus bedone:—

3 =4 + 4 + 44; 9 = 4 + 4 +44;

36 = 4(4 + 4) + 4; 45 = 44 +44;

52 = 44 + 4 + 4; 60 = 4 × 4 × 4 - 4.

This figure, which now forms a square, and the quarter of that square, can be so divided by two straight lines that its parts, separated and then reunited, form a perfect square. How is this done?

Squares

Solution

Here we have arranged five rows of five cards each, so that no two similar cards are in the same lines. Counting the ace as eleven, each row, column, and diagonal adds up to exactly twenty-six.

Playing cards

After you have looked at this Magic Square, and set it out on the table, shuffle the cards, and try to re-arrange them so as to give the same results.

Cut a square of paper or cardboard into seven such pieces as are marked in this diagram.

Tangram

Can you rearrange them so that they form the figure 8.

Solution

There are several ways in which strange juggling with figures and numbers is to be done, but none is more curious thanthis:—

Ask someone, whose age you do not know, to write down secretly the date and month of his birth in figures, to multiply this by 2, to add 5, to multiply by 50, to add his age last birthday and 365. He then hands youthis total onlyfrom this you subtract 615. This reveals to you at a glance his age and birthday.

Thus, if he was born April 7 and is 23, 74 (the day and the month) × 2 = 148; 148 + 5 = 153; 153 × 50 = 7650; 7650 + 23 (his age) = 7673; and 7673 + 365 = 8038. If from this you subtract 615, you have 7423, which represents to you theseventh day of the fourth month, 23 years age! This rule works out correctly in all cases.

Cut out in stiff paper or cardboard two pieces of the shape and size of the small triangle, and four pieces of the shapes and sizes of the other three patterns—fourteen pieces in all.

Pieces

The puzzle is to fit these pieces together so that they form a perfect oblong.

Solution

Here is a string of sentences, which may be used as stimulating mental gymnastics when we leave the “Land of Nod.”

A sleeper runs on sleepers, and in this sleeper on sleepers sleepers sleep. As this sleeper carries its sleepers over the sleepers that are under the sleeper, a slack sleeper slips. This jars the sleeper and its sleepers, so that they slip and no longer sleep.

Clever calculation has established a fact which we shall not be able to verify by personal experience. Whatever else may happen, the first day of a century can never fall on Sunday, Wednesday, or Friday.

On this cross there are seventeen distinct and perfect squares marked out at their corners by asterisks.

Cross

How few, and which, of these can you remove, so that not a single perfect square remains?

Solution

On a globe 2 feet in diameter the Dead Sea appears but as a small coloured dot. If it were frozen over there would be standing room on its surface for the whole human race, allowing 6 square feet for each person; and if they were all suddenly engulfed, it would merely raise the level of the lake about 4 inches.

Here is quite a good exercise for ingenious brains and fingers. Cut a piece of stiff paper or cardboard into such a right-angled triangle as is shown below.

Cross

Can you divide this into only three pieces, which, when rearranged, will form the design given as No. 2?

Solution

I sent an order for dwarf roses to a famous nursery-garden, asking for a parcel of less than 100 plants, and stipulating that if I planted them 3 in a row there should be 1 over; if 4 in a row 2 over; if 5 in a row 3 over, and if 6 in a row 4 over, as a condition for payment.

The nurseryman was equal to the occasion, charged me for 58 trees, and duly received his cheque.

This puzzle is not so easy of solution as it may seem at first sight.

Take a counter, or a coin, and place it on one of the points; then push it across to the opposite point, and leave it there. Do this with a second counter or coin, starting on a vacant point, and continue this process until every point is covered, as we place the eighth counter or coin on the last point.

Star puzzle

Solution

“Write down,” said a schoolmaster, “the nine digits in such order that the first three shall be one third of the last three, and the central three the result of subtracting the first three from the last.”

The arrangement which satisfies these conditions is, 219, 438, 657.

There is a sum of money of such sort that its pounds, shillings, and pence, written down as one continuous number, represent exactly the number of farthings which it contains. What is it?

Solution

2. If on a level track a train, running all the time at 30 miles an hour, slips a carriage, which is uniformly retarded by the brakes, and which comes to rest in 200 yards, how far has the train itself then travelled?

Solution

A traveller said to the landlord of an inn, “Give me as much money as I have in my hand, and I will spend sixpence with you.” This was done, and the process was twice repeated, when the traveller had no money left. How much had he at first?

Solution

How can we obtain eleven by adding one-third of twelve to four-fifths of seven?

Solution

5. Can you replace the missing figures in this mutilated long division sum?

Solution

I buy as many heads of asparagus in the market as can be contained by a string 1 foot long. Next day I take a string 2 feet long, and buy as many as it will gird, offering double the price that I have given before. Was this a reasonable offer?

Solution

7. As “one good turn deserves another,” first reverse me, then reverse my square, then my cube, then my fourth power. When all this is done no change has been made. What am I?

Solution

8. How can a thousand pounds be so conveniently stored in ten sealed bags that any sum in pounds from £1 to £1000 can be paid without breaking any of the seals?

Solution

This is at once a problem and apuzzle:—

Though you twist and turn me over,Yet no change can you discover.Take me thrice, and cut in twain,You will find but one remain.

Solution

Three gamblers, when they sit down to play, agree that the loser shall always double the sum of money that the other two have before them. After each of them has lost once, it is found that each has eight sovereigns on the table. How much money had each at starting?

Solution

11. When Tom’s back was turned, the boy sitting next to him rubbed out almost all hissum. Tom could not remember the multiplier, and only this remained on hisslate—

Can you reconstruct the sum?

Solution

12. The sum of the nine digits is 45. Can you hit upon other arrangements of 1, 2, 3, 4, 5, 6, 7, 8, 9, writing each of them once only, which will produce the same total. Of course fractions may be used.

Solution

13. The combined ages of Mary and Ann are forty-four years. Mary is twice as old as Ann was when Mary was half as old as Ann will be when Ann is three times as old as Mary was when Mary was three times as old as Ann. How old, then, is Mary?

Solution

Mr Oldboy was playing backgammon with his wife on the eve of his golden wedding, and could not make up his mind whether he should leave a blot where it could be taken up by an ace, or one which a tré would hit.

His grandson, at home from Cambridge for the Christmas vacation, solved the question for him easily. What was his decision?

Solution

15. I am aboard a steamer, anchored in a bay where the needle points due north, and exactly 1200 miles from the North Pole. If the course is perfectly clear, and I steam continuously at the rate of 20 miles an hour, always steering north by the compass needle, how long will it take me to reach the North Pole?

Solution

Three persons,A,B, andC, share twenty-one wine casks of equal capacity, of which seven are full, seven are half full, and seven are empty. How can these be so apportioned that each person shall have an equal number of casks, and an equal quantity of wine, without transferring any of it from cask to cask?

Solution

A hungry mouse, in search of provender, came upon a box containing ears of corn. He could carry three ears home at a time, and only had opportunity to make fourteen journeys to and fro. How many ears could he add to his store?

Solution

Take exactly equal quantities of lard and butter; mix a small piece of the butter intimately with all the lard. From this blend take a piece just as large as the fragment removed from the butter, and mix this thoroughly with the butter. Is there now more lard in the butter or more butter in the lard?

Solution

19. Here is an apparent proof that 2 =3:—

4 - 10 = 9 - 15

4 - 10 +25⁄4= 9 - 15 +25⁄4

and the square roots ofthese:—

2 -5⁄2= 3 -5⁄2

therefore 2 = 3.

Solution

20. Our Parcel Post regulations limit the length of a parcel to 3 feet 6 inches, and the length and girth combined to 6 feet. What is the largest parcel of any size that can be sent through the post under these conditions?

Solution

How can we show, or seem to show, that either four, five, or six nines amount to one hundred?

Solution

22. Can you arrange nine numbers in the nine cells of a square, the largest number 100, and the least 1, so that the product by multiplication of each row, column and diagonal is 1000?

Solution

23. Seven London boys were at the seaside for a week’s holiday, and during the six week-days they caught four fine crabs in pools under the rocks, when the tide was out at Beachy Head.

Hearing of this, the twenty-one boys of a school in the neighbourhood determined to explore the pools; but with the same rate of success they only caught one large crab. For how long were they busy searching under the seaweed?

Solution

24. On how many nights could a watch be set of a different trio from a company of fifteen soldiers, and how often on these terms could one of them, John Pipeclay, be included?

Solution

25. If Augustus Cæsar was born on September 23rd,B.C.63, on what day and in what year did he celebrate his sixty-third birthday, and by what five-letter symbol can we express the date?

Solution

Awas born in 1847,Bin 1874. In what years have the same two digits served to express the ages of both, if they are still living?

Solution

27.

A hundred and one by fifty divide,To this let a cypher be duly applied;And when the result you can rightly divine,You find that its value is just one in nine.

Solution

28. A man started one Monday morning with money in his purse to buy goods in a neighbouring town. He paid a penny to cross the ferry, spent half of the money he then had, and paid another penny at the ferry on his return home.

He did exactly the same for the next five days, and on Saturday evening reached home with just one penny in his pocket. How much had he in his pocket on Monday before he reached the ferry?

Solution

29. Three men gathered mangoes, and agreed that next day they would give one to their monkey and divide the rest equally. The first who arrived gave one to the monkey, and then took his proper share; the second came later and did likewise, and the third later still, neither knowing that any one had preceded him. Finally they met, and, as there were still mangoes, gave one to the monkey, and shared the rest equally. How many mangoes at least must there have been if all the divisions were accurate?

Solution

I look at my watch between four and five, and again between seven and eight. The hands have, I find, exactly changed places, so that the hour-hand is where the minute-hand was, and the minute-hand takes the place of the hour-hand. At what time did I first look at my watch?

Solution

31. There is a number consisting of twenty-two figures, of which the last is 7. If this is moved to the first place, the number is increased exactly sevenfold. Can you discover this lengthy number?

Solution

A farmer borrowed from a miller a sack of wheat, 4 feet long and 6 feet in circumference. He sent in repayment two sacks, each 4 feet long and 3 feet in circumference. Was the miller satisfied?

Solution

33. Five gamblers, whom we will callA,B,C,D, andE, play together, on the condition that after each hazard he who loses shall give to all the others as much as they then have in hand.

Each loses in turn, beginning withA, and when they leave the table each has the same sum in hand, thirty-two pounds. How much had each at first?

Solution

Knowing that the square of 87 is 7569, how can we rapidly, without multiplication, determine in succession the squares of 88, 89, and 90?

Solution

35.

Back to Index Next