New edition of the old hit!
OL GIC
S E L Z PUZ
BOOK
LOGIC
PUZZLES
BOOK #
CONTENTS I. THE MAGIC OF NUMBERS
...7
II. ILLUSTRATED PROBLEMS
...13
III. SOLVING THE SQUARE IV. LOGIC ABOVE ALL
...27
...37
V. PLAYING WITH MATCHSTICKS VI. BE OUR GUEST
...55
VII. ANSWER KEY
...63
...47
Šimon A. Đarmati BOOK #
LOGIC
PUZZLES #illustrations
LUKA DEJANOVIĆ
Hmmm… This is going
to be fun!
for Mila
PREDGOVOR Everyone has at least once been in a situation where you singlehandedly untangled a seriously tangled situation, a puzzling problem or a mind-bending riddle; found a key to the solution. Satisfied with yourself, you would then explain: ‘Oh, that’s nothing! I’ve got the brains, me!’ This statement does not only apply to the end result of the problem. Getting to it was also helped by your previous knowledge, but also skills that allowed you to utilize that knowledge. It was also important to correctly identify the problem, to spot where the issue lies, to figure out what’s the catch. After several somewhat unsuccessful attempts, different approaches, various paths towards the goal, as well as persistence or even audacity – why not? – you took a completely new road and reached success! Despite all your humility, you were, of course, visibly pleased with yourself. It is precisely this sense of satisfaction that was the catalyst for presenting you with this book. In fact, not just one book, but three of them, whose diverse activities will attract all those who are curious and persistent. Savour the joy of problem solving! Author
I THE MAGIC OF NUMBERS
LOGIC
PUZZLES
THE MAGIC OF NUMBERS
1. Guess the even number Have someone think of an even number, then multiply it by three, divide that product with two and multiply the quotient with three. After you find out the result, you will be able to guess the initial number. How? 2. Guess the odd number Have someone think of an odd number. Have them then multiply that number by three, add three to the product and then divide that sum by two and multiply the quotient by three. After you find out the result, you will be able to guess the initial number. How? 3. Guess the number Have someone think of a number and multiply it by three. Then ask them if the result is an odd or an even number. If it’s even, have that person divide the product by two and if it’s odd, have them add three to the product, then divide it
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by two. The result of the division should then be multiplied by three. After you find out the result, you will easily be able to guess the number. How? 4. Guess the two numbers Have someone think of two consecutive numbers lower than nine. Then ask them to multiply those two numbers with each other and subtract the smaller number from the product, then multiply that result with the smaller number. After they tell you the last digit of the result, you will easily be able to guess the numbers that person thought of. How? 5. Guessing small numbers Have someone think of a small number, multiply it with that same number, then add to the result the number they thought of multiplied by two, plus number one. After they tell you the result, you will be able to guess the initial number. How?
LOGIC
PUZZLES
THE MAGIC OF NUMBERS 6. How to find out a number Write down a number on a piece of paper whose digits produce a sum divisible by nine. Then turn around and ask the other person to multiply that number with a number of their choice. Then ask them to cross out one digit of the product that’s not zero, and write the remaining digits in a random order. After they tell you the result, you can guess which digit was crossed out. How? 7. The missing digit Have someone write down a number on a piece of paper. Add a digit to the left or right of that number so that the sum of digits in the new number is divisible by nine. After you’ve turned around, ask the other person to multiply that number with any natural number and leave out one digit from that product that’s not zero, then tell you the sum
of the remaining digits. When you hear the result, you will be able to guess the missing digit. Do you know how? 8. Quick addition Have someone write down several numbers with the same number of digits. You then write down several numbers, as well. After this, you will be able to quickly calculate the sum of all the written numbers. Which numbers should be written down and how can you find out the sum of all the numbers? 9. One for you and one for me Have someone write down three fivedigit numbers on a piece of paper. Take another piece of paper and write down a number, then say that you will write two additional numbers on the first piece of paper. The sum of these numbers and the ones already written will be equal to the number written on the second piece of paper. Which numbers do you need to write down and how do you get to the sum of those number? You can complicate things further by suggesting that, instead of three numbers, the other person writes even more numbers, but no more than nine. You can also write down the number on the second piece of paper after the other person has written their first number.
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LOGIC
PUZZLES
THE MAGIC OF NUMBERS
10. Guessing multiple numbers
12. Day of birth
Have someone think of several singledigit numbers and tell you how many numbers they thought of. Then ask them to multiply the first number they thought of by two and add five to that product. Then ask them to multiply that result by five and add ten, plus the second number they thought of, to the product. For the remaining numbers, the previous result should be multiplied by ten, then the following imagined number should be added to the product. After they tell you the end result, you will be able to guess the numbers that person thought of at the beginning. How?
Ask your friend to multiply the day of the month on which they were born with three, then divide the result by nine, multiply that result by three and divide the remainder (from the previous division) with three. Ask them to tell you the product and the quotient. Based on this information, you will be able to guess the day when your friend was born. How?
11. Guess the age Have a friend multiply their age by two, add four to that product, multiply that result by five, add twelve to that product, then multiply that number by ten. After you find out the end result, you will be able to guess how old your friend is. How?
13. Age difference Boast to a group of friends that you can guess the age difference between two people. The younger out of the two should subtract their age from 99, then the older one should add their age to the difference. After they tell you the result, you will easily be able to guess the age difference between the two of them. How? 14. Difference Ask someone to write down a double-digit number, then write the same number with swapped digits. Have them then subtract the smaller number from the greater one. Ask them to tell you the last digit of the difference. Based on this information, you will be able to the guess the difference between the two numbers. How? 15. Guess the quotient Have someone write down any triple-digit number in which the difference between
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LOGIC
PUZZLES
THE MAGIC OF NUMBERS the first and the last digit is equal to the number you will say. Have them then write down the number with the same digits, but switch the first and the last digit. Have them then subtract the smaller number from the greater one. The difference will always be divisible by nine, so you will be able to say the quotient of that difference and number nine. How? 16. Number 1089 You can make the previous problem even more interesting by writing down the number 1089 on a piece of paper, then place it in a sealed envelope. Give the envelope to the other person and ask them to write on it a triple-digit number in which the difference between the first and the last digit is greater than one. Then ask them to write the same number, but switch places of the first and the last digit, then subtract the smaller number from the greater one. Have them write down the same number as the difference, but switch the places of the first and the last digit, then add this number to the difference. Have them open the envelope with the piece of paper that has 1089 written on it. It will be equal to the sum derived by adding the difference and the number of the difference with swapped digits. How?
17. Imaginary number Have someone think of a number, then do the following: multiply that number by two, add five to the product, multiply the result by five, add ten to that product and multiply the sum by ten. Based on the given result, you will be able to guess the number they thought of. How? 18. Guess the odd number of numbers Have someone think of an odd number of numbers (three numbers, five numbers, seven numbers‌) and tell you the sum of the first and second number, sum of the second and third number, sum of the third and fourth number, etc. Finally, have them tell you the sum of the first and the last number. Based on this information, you will be able to guess the numbers they thought of. How? 19. Guess the even number of numbers Similar to the previous puzzle, have someone think of an even number of numbers (two numbers, four number, six numbers‌) and tell you the sum of the first and second number, sum of the second and third number, then sum of the third and fourth number, etc. Finally, have them tell you the sum of the first and the last number. Based on this information, you will be able to guess the numbers they thought of. How?
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LOGIC
PUZZLES
THE MAGIC OF NUMBERS
20. The magic table
5
4
3
2
1
16
8
4
2
1
There is a reason why this table is magic. Have someone think of a number no greater than 31 and tell you which columns contain this number. Without looking at the table, you will be able to guess this number easily. Work out the secret of the table and the way it was arranged.
17
9
5
3
3
18
10
6
6
5
21. Celebrity mathematician
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11
7
7
7
20
12
12
10
9
21
13
13
11
11
22
14
14
14
13
23
15
15
15
15
24
24
20
18
17
25
25
21
19
19
26
26
22
22
21
27
27
23
23
23
28
28
28
26
25
29
29
29
27
27
30
30
30
30
29
31
31
31
31
31
16
8
4
2
1
In this table, you will find numbers from 1 to 31 assorted in five columns based on a certain principle.
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Famous Hungarian mathematician Pataki posed the following problem before Anna, Bella and Klara. Anna has to think of a random even number and a random odd number, then tell one number to Bella and one to Klara. Bella is supposed to multiply her number by two and Klara her number by three. The products should then be added up and Bella and Klara should say the result out loud. Based on the result, Pataki knew to which girl Anna whispered the odd number and to which the even number. How?
II ILLUSTRATED PROBLEMS
LOGIC
PUZZLES
ILLUSTRATED PROBLEMS
1. Turning into a square
2. Cutting up five squares
Each of these eight figures can be turned into a square by making a single straight cut with scissors. The process has been shown in the picture, where figure A was turned into square C. However, that is only the eighth figure. How will you turn the rest into squares?
How can you cut five squares into three parts with three straight cuts so that three cut-outs can be used to form one square?
A
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B
C
LOGIC
PUZZLES
ILLUSTRATED PROBLEMS 3. Assemble a figure
4. Assemble a rhombus out of parts
Figures shown against the black background should be assembled using corresponding figures from the white backgrounds. Cutting the figures is not allowed; you should strive to solve this problem in your head. You can measure the figures, but only using your pencil.
Copy this hexagon onto a piece of paper, cut it into three parts, then use those parts to assemble a rhombus.
5. Assemble a square Copy this figure onto a piece of paper, cut it into four pieces with two cuts and use them to assemble a square.
6. Assemble a square out of a cross Copy the cross shown in the picture, then cut it as shown. Use the parts to form a square.
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LOGIC
PUZZLES
ILLUSTRATED PROBLEMS
7. Maintaining the balance
8. From point A to point B
Which two objects (two fish, two balls, two bells, two bars or a combination of two different objects) could be placed instead of the question mark so that the construct shown in the picture remains in balance? The weight of strings, unlike those of the bars, should be disregarded. All bars shown in the picture have an equal mass.
You should get from point A to point B by following the signs on the road in their order (top to bottom), shown to the left of the famous Amanpur labyrinth.
B
A 9. Numbers instead of question marks In each of the A, B, C and D sectors, the numbers of the target have been written in a certain order. Which numbers should be written instead of question marks?
D
A 1
10
4
19
7
37 ? 8 2
3
18 2
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? 39 6 63 8
?
13
10
73 5 14 ?
2 16 4 5 6
C
B
LOGIC
PUZZLES
ILLUSTRATED PROBLEMS 10. The Eschnapur labyrinth In this Eschnapur labyrinth, dark circles represent walls and white circles represent corridors. The passages in the walls are marked with corresponding numbers. You need to get to the centre and collect 500 points along the way. You start from the outside without any points and you win as many points as written on the passages you go through.. 10
421
213
22
50
2
0
1 41
99
312
38
4
5
0
2
89
47
120
1
52
450
5
21
3
1
3
3
1
78
17 02 4 500 86 9 418 15
3
2
345
390
7
24
8
45
1
40
11
7
25
238
rity of the vault. How did the formation of sixteen guards made by the lieutenant and the general look like?
60
11. Guard formation Sixteen guards were to be positioned around a square-shaped vault. After thinking long and hard, the police captain decided put five guards on each side of the vault, as shown in the picture. The lieutenant soon showed up and, unhappy with the formation set by the captain, rearranged the guards so that there were six on each side. Then came the general, who got very angry at the lieutenant, and placed seven guards on each side to reinforce the secu-
3
1
12. Shelf with cupboards Ducky the Duck had a square-shaped shelf with nine cupboards in his cellar. The central cupboard in the middle was supposed to store empty bottles, while the 60 bottles of quality wine (harvest year 1963) were arranged so that there were six bottles in every corner cupboard and nine bottles in every middle cupboard. This way, there were 21 bottles on each side of the square, as you can see from the picture. 6
9
9 6
6 9
9
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LOGIC
PUZZLES
ILLUSTRATED PROBLEMS
Ducky’s servant noticed that he checked his bottles by making sure there were 21 bottles on each side of the shelf. The servant then took four bottles and rearranged the rest so there were still 21 bottles on every side. Ducky counted the bottles using his method (21 along the sides of the square) and confirmed everything was in order. The greedy servant then took another four bottles and rearranged the rest, so there were 21 bottles on each side once again. He repeated this act of theft for as long as it was possible. The question arises – how many times did he steal the bottles and how many of them did he steal in total?
not participate in the next count. After the hearty meal was finished, they proceeded according to their agreement and the last one to stay in the pub was the owner. From which person did the count start? If there had been four soldiers at each table, where should have the count begun so that the owner would again be the last one to stay?
13. At the Empty Pint pub At the Empty Pint pub on the outskirts of the village, there was a table was set against each wall. After a military exercise, hungry soldiers looking for a meal visited the pub. Twenty-one soldiers came to the pub in total. Their seating arrangement is shown in the picture. There were seven soldiers sat at each of the three tables, while the pub owner sat at the fourth table (soldiers and the owner are illustrated with lines). The soldiers made a deal with the owner: whoever stays in the pub the last will pick up the tab, with the following rules – every seventh person would be exempt from paying and people would be counted clockwise (including the owner); this person would then leave the pub and
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14. Cut one part out A figure is made up of three squares, just as shown in the picture. Copy it onto a separate piece of paper, then cut out one part which will, when added to the remainder of the figure, form a square with a square-shaped opening in the middle.
LOGIC
PUZZLES
ILLUSTRATED PROBLEMS 15. One carpet out of two
17. Horses on the chessboard
Two square-shaped carpets were sold at the store, one measuring 60 x 60 centimetres and the other 80 x 80 centimetres. The costumer who came in, however, was looking for one 100 x 100 centimetres carpet. In an attempt to satisfy the costumer, the shop assistant promised that he would divide each of the carpets into two (without damaging the square pattern) and use these parts to assemble a carpet according to the wishes of the costumer. How did he do this?
Four horse chess pieces have been placed on the chessboard, in the order shown in the picture. The board should be divided into four equal parts, each of which should have one horse chess piece.
18. Package 16. Carpet with roses A carpet with seven roses is shown in the picture. The carpet should be cut into seven pieces with three straight cuts, so that there is one rose on each piece.
This package can be tied in several different ways, three of which have been shown in the picture. Which method will require the shortest piece of rope to tie the package and which will require the longest one? (The measurements of the box correspond to the inequation a + b > 2c.) 2 3 1 c
a
b
BOOK #
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LOGIC
PUZZLES
ILLUSTRATED PROBLEMS
19. Trainers
21. Lifting weights
How do the laces on the left and the right trainer look like from the inside?
Which of the A, B, C and D weights will be lifted and which will be lowered if the handle is moved in the direction as shown in the picture?
20. Nine little circles Nine little circles comprise tops of four small and three large equilateral triangles. Write numbers from 1 to 9 in the circles so that the sum of numbers in the tops of triangles is the same.
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A
B
C
D
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