Using Elementary Operations
To see how good your brain is, let's first put it to work on problems that require only perseverance, patience, sharpness of mind, and the ability to add, subtract, multiply, and divide whole numbers.
1. OBSERVANT CHILDREN
A schoolboy and a schoolgirl have just completed some meteorological measurements. They are resting on a knoll. A freight train is passing, its locomotive fiercely fuming and huffing as it pulls the train up a slight incline. Along the railroad bed the wind is wafting evenly, without gusts.
"What wind speed did our measurements show?" the boy asked.
"Twenty miles per hour."
"That is enough to tell me the train's speed."
"Well now." The girl was dubious.
"All you have to do is watch the movement of the train a bit more closely."
The girl thought awhile and also figured it out.
What they saw was precisely what the artist has drawn. What was the train's speed?
2. THE STONE FLOWER
Do you remember the smart craftsman Danila from P. Bazhov's fairy tale, "The Stone Flower"?
They tell in the Urals that Danila, while still an apprentice, took semiprecious Ural stones and chiseled two flowers whose leaves, stems, and petals could be separated. From the parts of these flowers it was possible to make a circular disk.
Take a piece of paper or cardboard, copy Danila's flowers from the diagram, then cut out the petals, stems, and leaves and see if you can put them together to make a circle.
3. MOVING CHECKERS
Place 6 checkers on a table in a row, alternating them black, white, black, white, and so on, as shown.
Leave a vacant place large enough for 4 checkers on the left.
Move the checkers so that all the white ones will end on the left, followed by all the black ones. The checkers must be moved in pairs, taking 2 adjacent checkers at a time, without disturbing their order, and sliding them to a vacant place. To solve this problem, only three such moves are necessary.
The theme of this problem is further developed in Problems 94–97.
If no checkers are available, use coins, or cut pieces out of paper or cardboard.
4. THREE MOVES
Place three piles of matches on a table, one with 11 matches, the second with 7, and the third with 6. You are to move matches so that each pile holds 8 matches. You may add to any pile only as many matches as it already contains, and all the matches must come from one other pile. For example, if a pile holds 6 matches, you may add 6 to it, no more or less.
You have three moves.
How many different triangles are there in the figure?
6. THE GARDENER'S ROUTE
The diagram shows the plan of an apple orchard (each dot is an apple tree). The gardener started with the square containing a star, and he worked his way through all the squares, with or without apple trees, one after another. He never returned to a square previously occupied. He did not walk diagonally and he did not walk through the six shaded squares (which contain buildings). At the end of his route the gardener found himself on the starred square again.
Copy the diagram and see if you can trace the gardener's route.
7. FIVE APPLES
Five apples are in a basket. How do you divide them among five girls so that each girl gets an apple, but one apple remains in the basket?
8 DON'T THINK TOO LONG
How many cats are in a small room if in each of the four corners a cat is sitting, and opposite each cat there sit 3 cats, and at each cat's tail a cat is sitting?
9. DOWN AND UP
A boy presses a side of a blue pencil to a side of a yellow pencil, holding both pencils vertically. One inch of the pressed side of the blue pencil, measuring from its lower end, is smeared with paint. The yellow pencil is held steady while the boy slides the blue pencil down 1 inch, continuing to press it against the yellow one. He returns the blue pencil to its former position, then again slides it down 1 inch. He continues until he has lowered the blue pencil 5 times and raised it 5 times-10 moves in all.
Suppose that during this time the paint neither dries nor diminishes in quantity. How many inches of each pencil will be smeared with paint after the tenth move?
This problem was thought up by the mathematician Leonid Mikhailovich Rybakov while on his way home after a successful duck hunt. What led him to make up this puzzle is explained in the answer, but don't read it until you have solved the problem.
10. CROSSING A RIVER
A detachment of soldiers must cross a river. The bridge is broken, the river is deep. What to do? Suddenly the officer in charge spots 2 boys playing in a rowboat by the shore. The boat is so tiny, however, that it can only hold 2 boys or 1 soldier. Still, all the soldiers succeed in crossing the river in the boat. How?
Solve this problem either in your mind or practically — that is, by moving checkers, matches, or the like on a table across an imaginary river.
11. WOLF, GOAT, AND CABBAGE
This problem can be found in eighth-century writings.
A man has to take a wolf, a goat, and some cabbage across a river. His rowboat has enough room for the man plus either the wolf or the goat or the cabbage. If he takes the cabbage with him, the wolf will eat the goat. If he takes the wolf, the goat will eat the cabbage. Only when the man is present are the goat and the cabbage safe from their enemies. All the same, the man carries wolf, goat, and cabbage across the river.
12. ROLL THEM OUT
In a long, narrow chute there are 8 balls: 4 black ones on the left, and 4 white ones — slightly larger — on the right. In the middle of the chute there is a small niche that can hold 1 ball of either color. The chute's right end has an opening large enough for a black but not a white ball.
Roll all the black balls out of the chute. (No, you can't pick them up.)
13. REPAIRING A CHAIN
Do you know why the young craftsman in the picture is so deep in thought? He has 5 short pieces of chain that must be joined into a long chain. Should he open ring 3 (first operation), link it to ring 4 (second operation), then unfasten ring 6 and link it to ring 7, and so on? He could complete his task in 8 operations, but he wants to do it in 6. How does he do it?
14. CORRECT THE ERROR
With 12 matches form the "equation" shown.
The equation shows that 6 - 4 = 9. Correct it by shifting just 1 match.
15. FOUR OUT OF THREE (A JOKE)
Three matches are on a table. Without adding another, make 4 out of 3. You are not allowed to break the matches.
16. THREE AND TWO IS EIGHT (ANOTHER JOKE)
Place 3 matches on a table. Ask a friend to add 2 more matches to make 8.
17. THREE SQUARES
Take 8 small sticks (or matches), 4 of which are half the length of the other 4. Make three equal squares out of the 8 sticks (or matches).
18. HOW MANY ITEMS?
An item is made from lead blanks in a lathe shop. Each blank suffices for 1 item. Lead shavings accumulated from making 6 items can be melted and made into a blank. How many items can be made from 36 blanks?
19. ARRANGING FLAGS
Komsomol youths have built a small hydroelectric powerhouse. Preparing for its opening, young Communist boys and girls are decorating the powerhouse on all four sides with garlands, electric bulbs, and small flags. There are 12 flags.
At first they arrange the flags 4 to a side, as shown, but then they see that the flags can be arranged 5 or even 6 to a side. How?
20. TEN CHAIRS
In a rectangular dance hall, how do you place 10 chairs along the walls so that there are an equal number of chairs along each wall?
21. KEEP IT EVEN
Take 16 objects (pieces of paper, coins, plums, checkers) and put them in four rows of 4 each. Remove 6, leaving an even number of objects in each row and each column. (There are many solutions.)
22. A MAGIC TRIANGLE
I have placed the numbers 1,2, and 3 at the vertices of a triangle. Arrange 4, 5, 6, 7, 8, and 9 along the sides of the triangle so that the numbers along each side add to 17.
This is harder: without being told which numbers to place at the vertices, make a similar arrangement of the numbers from 1 through 9, adding to 20 along each side. (Several solutions are possible.)
23. GIRLS PLAYING BALL
Twelve girls in a circle began to toss a ball, each girl to her neighbor on the left. When the ball completed the circle, it was tossed in the opposite direction.
After a while one of the girls said: "Let's skip 1 girl as we toss the ball."
"But since there are 12 of us, half the girls will not be playing," Natasha objected.
"Well, let's skip 2 girls!"
"This would be even worse — only 4 would be playing. We should skip 4 girls — the fifth would catch it. There is no other combination."
"And if we skip 6?"
"It is the same as skipping 4, only the ball goes in the opposite direction," Natasha answered.
"And if we skip 10 girls each time, so that the eleventh girl catches it?"
"But we have already played that way," said Natasha.
They began to draw diagrams of every such way to toss the ball, and were soon convinced that Natasha was right. Besides skipping none, only skipping 4 (or its mirror image 6) let all the girls participate (see a in the picture).
If there had been 13 girls, the ball could have been tossed skipping 1 girl (b)y or 2 (c), or 3 (d), or 4 (e), without leaving any girls out. How about 5 and 6? Draw diagrams.
24. FOUR STRAIGHT LINES
Make a square with 9 dots as shown. Cross all the dots with 4 straight lines without taking your pencil off the paper.
25. GOATS FROM CABBAGE
Now, instead of joining points, separate all the goats from the cabbage in the picture by drawing 3 straight lines.
26. TWO TRAINS
A nonstop train leaves Moscow for Leningrad at 60 miles per hour. Another nonstop train leaves Leningrad for Moscow at 40 miles an hour.
How far apart are the trains 1 hour before they pass each other?
27. THE TIDE COMES IN (A JOKE)
Not far off shore a ship stands with a rope ladder hanging over her side. The rope has 10 rungs. The distance between each rung is 12 inches. The lowest rung touches the water. The ocean is calm. Because of the incoming tide, the surface of the water rises 4 inches per hour. How soon will the water cover the third rung from the top rung of the rope ladder?
28. A WATCH FACE
Can you divide the watch face with 2 straight lines so that the sums of the numbers in each part are equal?
Can you divide it into 6 parts so that each part contains 2 numbers and the six sums of 2 numbers are equal?
29. A BROKEN CLOCK FACE
In a museum I saw an old clock with Roman numerals. Instead of the familiar IV there was an old-fashioned IIII. Cracks had formed on the face and divided it into 4 parts. The picture shows unequal sums of the numbers in each part, ranging from 17 to 21.
Can you change one crack, leaving the others untouched, so that the sum of the numbers in each of 4 parts is 20?
(Hint: The crack, as changed, does not have to run through the center of the clock.)
30. THE WONDROUS CLOCK
A watchmaker was telephoned urgently to make a house call to replace the broken hands of a clock. He was sick, so he sent his apprentice.
The apprentice was thorough. When he finished inspecting the clock it was dark. Assuming his work was done, he hurriedly attached the new hands and set the clock by his pocket watch. It was six o'clock, so he set the big hand at 12 and the little hand at 6.
The apprentice returned, but soon the telephone rang. He picked up the receiver only to hear the client's angry voice:
"You didn't do the job right. The clock shows the wrong time."
Surprised, he hurried back to the client's house. He found the clock showing not much past eight. He handed his watch to the client, saying: "Check the time, please. Your clock is not off even by 1 second."
The client had to agree.
Early the next morning the client telephoned to say that the clock hands, apparently gone berserk, were moving around the clock at will. When the apprentice rushed over, the clock showed a little past seven. After checking with his watch, the apprentice got angry:
"You are making fun of me! Your clock shows the right time!"
Have you figured out what was going on?
31. THREE IN A ROW
On a table, arrange 9 buttons in a 3-by-3 square. When 2 or more buttons are in a straight line we will call it a row. Thus rows AB and CD have 3 buttons, and row EF has 2.
How many 3- and 2-button rows are there?
Now remove 3 buttons. Arrange the remaining 6 buttons in 3 rows so that each row contains 3 buttons. (Ignore the subsidiary 2-button rows this time.)
32. TEN ROWS
It is easy to arrange 16 checkers in 10 rows of 4 checkers each, but harder to arrange 9 checkers in 6 rows of 3 checkers each. Do both.
33. PATTERN OF COINS
Take a sheet of paper, copy the diagram on it, enlarging it two or three times, and have ready 17 coins:
20 kopeks 5 15 kopeks 3 10 kopeks 3 5 kopeks 6
Place a coin in each square so that the number of kopeks along each straight line is 55.
[This problem cannot be translated into United States coinage, but you can work on it by writing the kopek values on pieces of paper — M.G.]
34. FROM 1 THROUGH 19
Write the numbers from 1 through 19 in the circles so that the numbers in every 3 circles on a straight line total 30.
35. SPEEDILY YET CAUTIOUSLY
The title of the problem tells you how to approach these four questions.
(A) A bus leaves Moscow for Tula at noon. An hour later a cyclist leaves Tula for Moscow, moving, of course, slower than the bus. When bus and bicycle meet, which of the two will be farther from Moscow?
(B) Which is worth more: a pound of $10 gold pieces or half a pound of $20 gold pieces?
(C) At six o'clock the wall clock struck 6 times. Checking with my watch, I noticed that the time between the first and last strokes was 30 seconds. How long will the clock take to strike 12 at midnight?
(D) Three swallows fly outward from a point. When will they all be on the same plane in space?
Now check the Answers. Did you fall into any of the traps which lurk in these simple problems?
The attraction of such problems is that they keep you on your toes and teach you to think cautiously.
36. A CRAYFISH FULL OF FIGURES
The crayfish is made of 17 numbered pieces. Copy them on a sheet of paper and cut them out.
Using all the pieces, make a circle and, by its side, a square.
37. THE PRICE OF A BOOK
A book costs $1 plus half its price. How much does it cost?
38. THE RESTLESS FLY
Two cyclists began a training run simultaneously, one starting from Moscow, the other from Simferopol.
When the riders were 180 miles apart, a fly took an interest. Starting on one cyclist's shoulder, the fly flew ahead to meet the other cyclist. On reaching the latter, the fly at once turned back.
The restless fly continued to shuttle back and forth until the pair met; then it settled on the nose of one of the cyclists.
The fly's speed was 30 miles per hour. Each cyclist's speed was 15 miles per hour.
How many miles did the fly travel?