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.
What is the secret in the pattern of this stained glass?
SOLUTION
The image is a superposition of a blue shape and a yellow shape. The places where they coincide are colored in green (blue + yellow = green). The blue shape is consisting of horizontal stripes with lengths 3, 1, 4, 1, 5, 9, 2, 6, 5, representing the number pi, and the yellow shape is consisting of vertical stripes with lengths 4, 6, 6, 9, 2, 0, 1, 6, 1, representing the Feigenbaum constant.
A man must mail a precious necklace to his wife, but anything sent through the mail will be stolen unless it is sent in a padlocked box. A box can bear any number of padlocks, but neither of the spouses has the key to a lock owned by the other. How can the husband mail the necklace safely to his wife?
SOLUTION
The man can put a lock on the box and send it to his wife. Then she can put her own lock and send it back. Once the man receives the box, he can remove his lock and send the box once again to his wife. When she gets it, she can finally unlock the box using her own key.
On the picture, you can see an example of a wall made of 2×1 bricks. On the wall, there are 2 cracks, which are straight lines passing through the whole wall from top to bottom and from left to right, without intersecting any bricks.
Can you make the following walls without any cracks:
wall 5×6 with 15 bricks;
wall 6×6 with 18 bricks?
SOLUTION
The solution for a 5×6 wall is shown below. However, if the wall has dimensions 6×6, it is impossible to build it without any cracks. Indeed, assume the wall does not have any cracks. Therefore every line passing through it must intersect 2, 4, or 6 bricks. Since there are in total 10 lines passing through the wall and each brick is intersected by exactly one of them, the total number of bricks must be at least 10 x 2 = 20 > 18. This yields a contradiction.
A large rectangle is partitioned into smaller rectangles, each of which has integer length or integer width. Prove that the large rectangle also has integer length or integer width.
SOLUTION
This problem can be solved using graph theory, but the most elegant solution is based on some basic calculus.
Place the big rectangle in the plane so that its sides are parallel to the X and Y axes. Now integrate the function f(x)=sin(πx)sin(πy) over the boundary of any small rectangle. Since at least one of its sides has integer length, the result will be 0. If you sum all integrals taken over the boundaries of the small rectangles and cancel the opposite terms, you will get that the integral of f(x) over the boundary of the large rectangle is also equal to 0. Therefore at least one of its sides has integer length.
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