There is an island on a planet and infinitely many planes on it. You need to make one of these planes fly all around the world and land back to the island. However, each of the planes can carry fuel which is enough to travel just half of the way, and fuel cannot be stored anywhere on the planet, except for the island and in the planes. Assuming the planes can refuel each other mid-air, and all of them must eventually arrive safely back on the island, how many of them do you need to accomplish the task?
SOLUTION
3 planes are enough, label them A, B, C. They leave the island simultaneously in a clockwise direction, and after 1/8 of the way, A refuels B and C completely, then turns back towards the island. B and C continue to fly until they get to 1/4 of the way, where B refuels C completely and turns back towards the island. When C gets mid-way, A and B leave the island counter-clockwise, and after 1/8 of the way, A refuels B completely and turns back towards the island. B continues towards C, and when the two planes meet, they share their fuel, then fly together towards the island. In the meantime A arrives on the island, refuels completely, and starts flying again counter-clockwise towards B and C, so that it can meet them and give them enough fuel, so that all of them arrive safely on the island. It is easy to see that 2 planes are not enough.
A mother has 4 children. On Halloween, she decides to give them 200 candies one by one, starting with the youngest child, then the second youngest, third youngest, etc. At some point, they have to interrupt the procedure and eat dinner, but when they continue, nobody remembers which was the last child to get a candy. How can the mother distribute the remaining candies to the kids without starting over again or counting how many candies have been given so far?
SOLUTION
The mother can start giving the remaining candies one by one in the reverse order, starting with the oldest child, then the second oldest, etc.
Can you draw a 3-dimensional object which looks like a square when looked from one side, a triangle when looked from a second side, and a circle when looked from a third side?
Move just two matchsticks so that you get as large a number as possible.
SOLUTION
You can take the two horizontal matchsticks from the first 0, form the number 11 with them to get 1100011. You can also make 1111000 if you don’t mind that the power’s font is larger than the base’s font, or use the quite uncommon operation “tetration” to get 1111000.
A dog is standing with three of its legs on a measuring scale, keeping its fourth leg in the air. If the scale is displaying 30lbs, how much will it display if the dog places its fourth leg on the scale?
SOLUTION
The answer is 30lbs since it does not matter whether the fourth leg is in the air or not.
If you hang a painting over two nails on the wall and remove one of the nails, the string of the painting will stay over the other nail, and the painting will keep hanging. Can you wrap a string around the nails in such a way, so that after removing any of the nails, the painting will fall?
There are a fly and three spiders on the edges of a cube. If the spiders’ velocities are at least 1/3 of the fly’s velocity, and all insects can travel only along the edges of the cube, show that the spiders can eventually catch the fly.
SOLUTION
Choose two opposite edges of the cube, then let two of the spiders “protect” them from the fly. You can do this by keeping the distance from the two spiders to the edges’ endpoints at most 1/3 of the distance from the fly to these endpoints. The remaining parts of the cube do not contain any loops, so the third spider can easily catch the fly there.
How many places are there on Earth so that if you travel 1 mile South, followed by 1 mile East, followed by 1 mile North, you will get back where you started from?
Remark: You can assume Earth is a perfect globe.
SOLUTION
The answer is infinitely many. Of course, the North Pole is one such place. However, if you start close to the South Pole, such that after traveling 1 mile South you land on a parallel with total length of 1/N miles, N-integer number, then when traveling East you will encompass this parallel exactly N times and later will get back to the starting place. These are all places with the property described above.
A few days ago, my girlfriend broke up with me. We were never really right for one another, and I knew it was coming, but of course, it still stung a bit. And true to her style, she did it via a series of poorly written text messages. She didn’t really pull any punches, mentioning many things she saw as my character flaws: playing too many video games, considering a trip to Taco Bell a “date”, constantly correcting her grammar and spelling, shaving my chest with her razor, shaving my chest at all, etc.
At first I didn’t know how to respond, but eventually I decided to simultaneously take the high road and the low road. So I sent her this email:
Baby,
I’m sad your leaving.
I can’t believe we ended up here, after what was an wonderful beginning. I know at times things were good and at times they were badly, but I always thought we would make it.
I remember the start. Right when I saw you, I knew you were a people of interest. Things were great for so long. But now, we all know whom is at fault, so I understand your decision.
The dogs are upset. Sandy just lays down looking sad and misses u. So does Rusty. They makes me feel better at least. I’m glad the shelter gave them to you and I.
I feel nauseous without you—I feel like I could throw up any second—but I understand. This is the way things had to be.
Hears what I want you too do: live your life, be happy, and be goode.
By 4ever, –Dan
And I smiled smugly to myself. I knew that she would never understand what I was really saying. Do you?
SOLUTION
In the text there are some words which are written incorrectly:
Baby,
I’m sad your leaving.
I can’t believe we ended up here, after what was an wonderful beginning. I know at times things were good and at times they were badly, but I always thought we would make it.
I remember the start. Right when I saw you, I knew you were a people of interest. Things were great for so long. But now, we all know whom is at fault, so I understand your decision.
The dogs are upset. Sandy just lays down looking sad and misses u. So does Rusty. They makes me feel better at least. I’m glad the shelter gave them to you and I.
I feel nauseous without you—I feel like I could throw up any second—but I understand. This is the way things had to be.
Hears what I want you too do: live your life, be happy, and be goode. By 4ever,
–Dan
If you take these words, and correct the mistakes, you will get: You’re a bad person who lies. You make me nauseated. Here’s to good bye forever.
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