A town consists of 3 horizontal and 3 vertical roads, separated by 4 square blocks. A policeman and a thief are running along the roads with speeds of 21km/h and 10km/h respectively. Show that the policeman has a strategy ensuring he will eventually see the thief.
Remark: The policeman can see the thief if they are on the same road at some moment. He has no idea about his position at any time.
SOLUTION
A working strategy for the policeman would be to go to the center and to start encompassing the four blocks clock-wise one by one, in a clockwise manner.
Since the policeman is twice as fast as the thief if the thief is in the center of the town at some point, then there exists a moment in which the policeman is in the center, and the thief is not on the boundary, i.e. he gets shot.
Now, assuming the thief never visits the center, his angle with respect to the coordinate system defined by the two middle roads changes continuously. The angle of the policeman with respect to the same coordinate system can be defined to change continuously as well. Since the policeman needs less time to increase his angle with 360 degrees than the thief, there will be a moment when the two have the same angle. However, this implies that the policeman will be able to see the thief and shoot him.
One early morning, a group of friends meets in their favorite café.
Ash, the biggest in the group, remarked :
“Must have been millions of years since we were all together, uh?”
Affie, who was wearing her brand new elephant pants, nodded. Anthony kept on complaining about the weather back home.
“Yeah, I am glad to be here with you, it is so cold at my place!”
Eugenie, who was saddened by the loss of a friend, was looking at Samuel and Namur, who were arguing about the possible future election of Donald Trump.
“These two are really inseparable”, she said to herself.
Octavia, the smallest among her friends, stood and spoke :
“Guys, I have a surprise for you! We’re going to the opera tonight!”
The waiter, waiting for the orders, wondered why these customers reminded him of something. But all of a sudden, he said:
“Ladies and gentlemen, may I suggest some sliced bread with butter, slices of cheese and ham? We also have croissants and other pastries. And for drinking, is coffee fine? We also have tea, of course, and orange and apple juice for you.”
Which is this group of friends, and what came to the waiter’s mind?
SOLUTION
These are the seven continents – Ash (Asia), Affie (Africa), Anthony (Antarctica), Eugenie (Europe), Samuel (South America), Namur (North America), Octavia (Oceania). Their entire conversation consists of various hints. In the end, the waiter was thinking about bringing them Continental Breakfast.
Leonid Mochalov is a Russian puzzle writer and inventor, who has published several brain teaser books and patented numerous mechanical puzzle toys. We decided to invite him for an interview at Puzzle Prime and learn about his passions, habits, and views on Pokemon GO. You can see all of Leonid’s puzzles on his personal website.
Interview
Q. Hello Leonid. When did you discover your passion about puzzles? Was it influenced somehow by your education or occupation at that time?
A. It happened during my school years. I published my first puzzle when I was 13 years old. Education and occupation had nothing to do with it.
Q. What drives you to make all these puzzles?
A. Ambitions. When I was young I decided to become a famous puzzle creator, such as the great Henry Dudeney or Sam Loyd.
Q. You have invented many mechanical puzzles. Do you ask professionals to build them based on your concepts, or you do it yourself?
A. I post the blueprints of my puzzles and their descriptions on my website for free. After that, various puzzle collectors and companies make physical copies of them.
Leonid’s puzzle blueprints
Q. Designing 3-dimensional puzzles seems incredibly challenging. Can you describe how the process of creating one goes?
A. I have a very well developed 3D imagination. First, I think of a geometric object of an unusual shape, and divide it into smaller pieces. Then the pieces are glued together and the brute-force method comes. Another possible approach is to come up with a single piece which locks the entire structure.
Q. Which is your favorite physical puzzle of all time?
Q. And which is your favorite puzzle you have invented?
A. The “Pyramid” – it is an analogue of the “Soma Cube”.
By the way, despite the abundance of mechanical puzzles in stores, the real (classical) ones are not more than 100. These two can be classified as such.
Q. Do you like more mechanical puzzles or pen and paper problems?
A. I lost my interest in pen and paper problems in the early 90s, and then I focused my interest on mechanical puzzles. I believe a good mechanical puzzle is worth a lot more than any pen and paper problem.
Place the numbers from 1 to 9 in the squared so that all the equations are accurate.
Q. Where can people buy your puzzles from?
A. Many of my mechanical puzzles are sold in specialized web-stores.
Q. It looks like the new generations do not appreciate problem-solving as much. Do you think there is still appreciation for puzzles nowadays?
A. Not that much in our country. The pick of the interest was in the last quarter of 20th century. After that computerization and shooting games came into play. The contemporary epidemic is “Tank Wars” and “Pokemon Go”.
Q. Do you keep in touch with other inventors, maybe collaborate on projects together?
A. Not quite, I tend to work mostly alone.
Q. Some people find listening to music during work quite helpful. What about you – do you listen to anything while designing your puzzles?
A. I don’t listen to music when I work, because I find it distracting. However, I listen to disco style songs of 60-70s in my spare time.
Q. You seem a bit old-school. Do you work often on computer, play video games, watch movies? What are your hobbies overall?
A. I do non-professional photo and video shooting, so sometimes it is necessary to work with materials on my computer. Other activities I practice are following news, downloading movies and programs, watching Sci-Fi’s and thrillers. I don’t play video games however. I also used to play card games like “Preference” before retirement.
Leonid fishing
Q. You look in good shape; we know you enjoy fishing. Do you do any other sports?
A. I don’t actually. I used to jog, but then gave it up.
Q. Do you think puzzle solving is beneficial for people, especially younger children and teenagers?
A. It is very necessary for development of young generations. Puzzle solving is educational and distracts from drugs and alcohol.
Q. Tell us a funny science joke or an anecdote.
A. Something I find amusing is related to Einstein’s had 2 cats. It is common in the West people to have an opening in the door for their pets. However, Einstein had two openings – one for the big cat and the other for the small cat. It wouldn’t be like that in case of a normal person since the small cat can go through the big hole too.
Q. Thank you for the interview Leonid, wish you all the best in your endeavors.
You have a jar filled with water and a glass. If you pour some water into the glass and place a cork in it, the cork will float towards the edges of the glass. What is the easiest way to make the cork float towards the center?
SOLUTION
Since the liquid molecules adhere to the glass molecules on the sides of the glass, the water level there is higher and buoyancy makes the cork float in that direction. If you fill the glass all the way to the edge, then the water surface will be convex and the cork will float towards the center.
Look at the picture below and answer the following questions:
What time of the day is it?
Is it early spring or a late fall?
Is the river navigable?
Which direction does the river flow? (North, East, South or West?)
Is the river deep or shallow at the side where the boat is?
Is there a bridge across the river nearby?
How far is the railroad from here?
Do the birds fly North or South?
SOLUTION
2. People are sowing the crops, so it is early spring. 8. Since it is spring, birds are flying North. 1. Since birds are flying North, the shadows are pointing East, and therefore it is morning. 4. Judging by the water around the buoy, the river is flowing South. 3. Since there is a buoy, the river must be navigable. 5. The fishing line is long, so the river must be deep. 6. There would not be a ferry if there was a bridge nearby. 7. The guy on the left looks like a railroad worker, so probably the railroad is nearby.
This thing all things devours, Birds, beasts, trees, flowers, Gnaws iron, bites steel, Grinds hard stones to meal, Slays king, ruins town, And beats mountain down.
The following position occurs in a real game, right after one of the pieces gets knocked off the board. What was the piece?
SOLUTION
It was a black knight. First, notice that the black pawns have moved 14 times diagonally and thus they have taken 14 pieces. Therefore the knocked off piece is black. Since it is impossible for both kings to be checked at the same time, the missing piece was positioned on a2. It couldn’t be a queen or a rook, because the white king would be checked both by it and the pawn on b3, which is impossible. Therefore the missing piece is either the black white-squared bishop or the black knight. However, the pawns on b7 and d7 haven’t been moved the entire game and then the black white-squared bishop hasn’t either. Thus we conclude that the knocked off piece is a black knight.
