Test Your Wits on These Mathematical Puzzles (Mar, 1932)
The Four Color Theorem was not proven until 1976 and required the use of a computer.
I’m pretty sure the thing about arabic numerals representing the number of angles in their characters is total B.S.
Test Your Wits on These Mathematical Puzzles
by WILLIAM J. HARRIS
There’s nothing like a puzzle to test one’s mental alertness, and those presented here by Mr. Harris are certainly corkers. He also gives you some simple tricks which, though they only take a few minutes to learn, will convince your friends that you are a mathematical wizard of the first water. (P. S.— Answers are in the back of the book!)
FIFTEEN Christians and fifteen Turks once were caught at sea in a terrific storm, according to an old story popular way back in the Middle Ages. The captain said half the passengers would have to be tossed overboard to lighten the ship. A Christian proposed they form a circle, and, starting from a given point, every ninth man be counted out and thrown overboard. When the operation was complete there were 15 Christians left, and every Turk had been drowned.
Could you put thirty counters in a circle and figure out the arrangement to produce that result?
Mathematical magic has intrigued countless generations since men first began to count. The magical properties of numbers for ages caused them to be regarded with superstition, and, since education has abated that awe, the interest continues. Take the familiar trick problem of determining a person’s age, or determining a number of which the person is thinking. There are literally dozens of ways to do the stunt, and all of them baffling to the uninitiated.
There is a story told of the Seven Wise Men of Gotham, who taught a school of 24 pupils. They built a schoolhouse of eight rooms, arranged in a square about an open court, and placed 3 pupils in each room. Attendance could be checked by adding up the pupils on each side, the correct number being nine, as three rooms were visible to a side. But four pupils slipped out, and the others rearranged themselves, so with only 20 present there were still nine to the side. Then the four came back and brought four friends, but, with another arrangement, there were still nine to the side, though 28 were present. And soon four more strangers slipped in, but there were still 9 to a side, despite the fact the school now housed 32. Can you figure out the various arrangements?
That’s something of a magic square. Not a true magic square, because in those the sums of all vertical and horizontal columns, and likewise the diagonals, are the same. And some of them have many other queer properties.
Benjamin Franklin, who produced a sixteen sided magic square which he described as the “most magically magical of any magic square ever made by any magician,” boasted that in addition to the usual properties, all bent lines of 16 cells totaled the same as straight rows, and if a four by four window was cut in a sheet of paper, so that just sixteen of the cells would be visible through it, the sixteen, no matter where the window was placed, likewise would total the same as a row.
The Mystery of Numbers
The making of magic squares was once a mystical art, and many volumes have been written about the various systems. The smallest possible magic square is one of 3 by 3 cells, with nine consecutive numbers. There is just one possible way to arrange nine numbers to form a perfect 3 by 3 square, and if you memorize that arrangement you can write a square instantly with any nine consecutive numbers, regardless of their size.
Of course, in addition to the one way, there are eight different ways of presenting it, depending on which of the four sides is uppermost and whether the numbers are written from left to right or right to left.
From that single 3 by 3 square the jump to 4 by 4 is enormous, for there are no less than 4,352 ways of producing the latter. And when you go to 5 by 5, with 25 cells, there are at least 28,800 ways of arranging the numbers.
The supposed origin of the Arabic numerals we use today is itself an interesting example of order. They are believed to descend from collections of angles, each group totaling the number it represented. The 1 was one angle, formed by the vertical shaft and the short slanting bar at the top, while the angular 2, made like a “Z”, showed two angles. The Germans still write the figure 7 with the cross bar needed to provide its four extra angles. The zero, being oval, has no angles, of course.
A Number-Finding Trick
Finding a number selected by someone can be done in many ways. Here’s an easy one to remember: Ask the person choosing the number to treble it. Enquire if the product is even or odd. If it is even ask him to take half of it, if odd to add one and then divide by 2. Tell him to multiply the half by 3. Ask how many integral times 9 divides into the product. If the answer is 3, for example, the number thought of was either 2 times 3 or 2 times 3 plus one, according to whether the result of the first step was odd or even.
Arithmetical progressions are always a surprise to the uninitiated. For example, the old one about getting rich in one month. All you need do is save one cent the first day, two cents the second, four the third,
and so on, doubling the amount of your savings each day. Of course, toward the end of the month it gets rather hard, particularly on the last day, when you must save more than $5,000,000. But then your total fortune after 30 days will be more than ten and three-quarters millions.
A variation of that one is to offer to work for one year at a salary of 1 cent the first week, 2 cents the second, and so on. That appeals to some employers, but always ask for a bond to insure payment of the last week’s salary. For that final week’s pay will be some 1,500 times all the money in the world, and the year’s earnings will come to some 40,000 billion dollars.
By the way, there is an easy way to get the total of any series of progressions such as the above. Just multiply the last number by 2 and subtract the first. For example, if your series is 1, 2, 4, 8, 16, 32, 04, 128 they total 255, while twice 128 is 250 and subtracting the first number gives 255.
The “Take-Away” Stunt
A good parlor game with checkers, matches or other counters is the take-away game. You place 15 counters in a pile and invite an opponent to alternate with you in taking away one, two or three at a time, the object being to force the loser to take the last counter. The trick is to leave five on your next to the last draw, then you can force him to take the final one.
A companion of the story of the Turks and the Christians is the equally ancient one of the Arab who died and left 17 camels to his 3 sons. His will stipulated that the eldest son should have one-half the estate, the second son one-third and the third son one-ninth. As 17 is not divisible by 2, 3, or 9, the sons borrowed a camel from a neighbor, put it with their father’s 17, and proceeded to divide. The eldest son took 9 for his half, the second son got 6 for his third, and the third son 2 for his ninth, 9, 6 and 2 totaling 17, whereupon they returned the borrowed beast to its owner. A little calculation will show the eldest son was entitled to 8-1/2 camels, but got 9, so he profited by the deal. The second son was entitled to 5-2/3 camels, but got 6, so he profited. And the third son was entitled to 1 8/9 camels, but got 2, so all three actually got more than their share.
The solution, of course, is that 1/2 plus 1/3 plus 1/9 only totals 17/18, or less than their father’s whole fortune.
Why Does a Map Need But Four Colors?
There’s one mathematical problem, the correct answer of which has been known for generations, but for which no absolute proof has ever been found. That’s the question of why a map maker needs but four colors to picture the territories on any possible map, so that no two adjacent sections will have the same color.
Map makers had known that for a long time before science first heard of the problem a century ago. Since that time some of the world’s best mathematical brains, including the great De Morgan, have wrestled with it in vain. They know it is so, but can’t find a scientific proof that fits all cases.
A Square Inch From Nowhere
Here’s another puzzle which has mystified people for generations. A ship has a hole in the bottom measuring 5 by 13 inches, and the only available board to patch it is 8 inches square. The hole embraces 65 square inches, the board contains but 64. But the ship’s carpenter knows how to saw up a 64 square inch board to get 65 square inches out of it! One of the illustrations shows how he did it.
Of course, the explanation is that the parts when cut up and refitted do not actually lit. In between there is a crack, diamond shaped, as long as the diagonal, and extremely narrow, which is not visible when you cut the parts up and lay them down in the new arrangement. Lay out a square of eight rows to a side, cut it and see for yourself.
Something for Nothing
A variation of the same trick is to lay out a sheet of 11 by 13 inches, or units, and cut it diagonally. Move one section upward one square, and you apparently have your original 143 square inches plus two triangles equal to one half a square each, or a total of 144 squares. Move it another square, and you seem to have 143 squares plus two full and two half squares. You can keep on moving it, with apparent gains, but you soon discover that, the original not having been a square, the cut was not at 45 degrees, and so you no longer have a rectangle, for the parts don’t fit.
Just the other day a friend who is a railroad man submitted a railroad switching problem to see if I could solve it. The problem was well known in England about 40 years ago but appears to have recently been revived over here. In the problem a “Y” extends from the main line, the leg of the “Y” being so short that only one car can be placed on it at a time. There is a car spotted on each arm of the “Y” and it is the task of the switching engine to reverse their position, putting car “A” where car “B” has been and “B” in place of “A”. The diagram shows the arrangement. The solution is simple for any accomplished railroad man.
The Wasp and the Fly.
In a room 30 feet long, 12 feet wide and 12 feet high a wasp is clinging to the end wall, on the middle line, one foot from the ceiling. On the middle line of the other end wall, and 11 feet from the ceiling is a fly. The wasp crawls over and eats the fly.’ What’s his shortest path?
If you want to solve that one lay out the room on paper in such a way that you can draw a straight line from wasp to fly.
There was a time when the great mathematicians loved to play with problems like these, and expound the answers before scientific bodies. Euler, for example, is still famous as the man who solved the problem of the seven bridges of Koenigsburg in his memoir before the St. Petersburg Academy of Russia in 1736. The town, near the mouth of the river Pregel, includes the island of Kneiphof, and the river was spanned by seven bridges. Someone asked whether it; was possible, in one walk, to cross all seven bridges once and none of them twice.
Euler proved it was possible, and laid down the rule that a unicursal figure—one with no open end—could not be traced at one stroke when there were more than two points of odd order, that is, points from which an odd number of paths diverged. And when the Koenigsburg bridge problem is reduced to a geometrical design it has four points of odd order.
From the old mathematical problems of the ancients to lightning calculation is quite a jump, but if you would like to establish a reputation in the latter field a few simple tricks will be of big aid. The true lightning calculator, by the way, is usually a child prodigy who outgrows the gift as he grows up, and who seldom has been able to explain his mental processes. But with some simple rules memorized you can present a number of baffling tricks.
Consider this one: You can tell a friend to choose any number consisting of from one to nine consecutive digits—a number like 123,456 for example—have him multiply it by nine, add a number one greater than the final digit—in this case add 7—and you can give the answer instantly. It’s simple, for in that particular multiplication table of nine all answers consist of ones, and the answer will have one more digit than the chosen number, in other words if you multiply 123,456 by 9 and add 7 the answer is 1,111,111.
Recently a new twist to an old trick has spread like wildfire at parties in Chicago. The victim is invited to select any number of three digits, in which the difference between the first and last digits exceeds one. Form a new number by reversing the first one, and subtract one from the other. Reverse the answer, and add the two. To that add 65.
The victim is then told to take the Chicago telephone directory, look on the page of that number, where he will find the name of his club listed as the second entry in the fifth column. The entry reads: “Royal Order of Bone Heads.”
The trick, minus the last addition of 65 and the part about the telephone book, is several centuries old. Up to that last step any numbers chosen will produce 1089 as an answer. Someone, seeing the telephone book listing and recalling the old mathematical trick, worked out the new presentation, in which the correct page, 1154, was reached by adding 65 to the 1089.
Bachet, in his classical “Problemes” proposed a problem in weights, asking how many scale
weights would be needed to weigh any integral number of pounds from one to forty. There are two answers, depending on whether all the weights must be placed in one pan, or whether they can be placed in both-
The first would take weights of 1, 2, 4, 8, 16 and 32 pounds. The second would require only 1, 3, 9, and 27 pounds. The one pound weight would weigh one pound in the other pan, two pounds would be weighed with the material and the one pound in one pan, and the three pound weight in the other, and so on.
The Seven Bridges of Koenigsburg
The great mathematician, Euler, reduced the problem of the seven bridges of Koenigs-burg to the above geometrical figure to prove the impossibility of crossing all bridges once without crossing any bridge twice. In the diagram, A, B, C, and D represent the land areas, and 1, m, n, p, q, r, and s the bridges.
Solution of the Turk and Christian puzzle:
Arrange the Turks and Christians in a circle in the following order: 4 Christians, 5 Turks, 2C, 1 T, 3 C, 1 T, 1 C, 2 T, 2 C, 3 T, 1 C, 2 T, 2 C, 1 T. Cast out every ninth man, until 15 are gone, and you’ll end up with all Christians.
The Wise Schoolmasters of Gotham:
This is the arrangement of the school, figures indicating the number of students in each room, totaling nine on each side of the square:
3 3 3
3 0 3
3 3 3
This is how four extra students were accommodated without the schoolmasters’ knowing it:
2 5 2
5 0 5
2 5 2
This is the arrangement of students when four of them played hookey:
4 1 4
1 0 1
4 1 4
And here’s how 32 students fitted into the rooms, while according to the master’s roll call there were only 24:
1 7 1
7 0 7
1 7 1