Take a look at the above picture, and agree with me that this is weird. We see two triangles, and based on the grid they are both the same size. A quick count of the grid boxes tells us that the surface of both triangles should be 13 × 5 ÷ 2. All the pieces are equal in size, yet by shuffling them around we suddenly have a spare grid box!
This does not make any sense. The act of rearranging the triangle pieces should not change the surface area. What's going on here?
Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that the ‘surprise hanging’ can't be on a Friday, as if he hasn't been hanged by Thursday, there is only one day left - and so it won't be a surprise if he's hanged on a Friday. Since the judge's sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday.
He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn't been hanged by Wednesday night, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. By similar reasoning he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all.
The next week, the executioner knocks on the prisoner's door at noon on Wednesday — which, despite all the above, will still be an utter surprise to him. Everything the judge said has come true.” source
An interesting paradox! There are many variations to this story. For example, another popular version is about a surprise examination in class. Resolutions from varying perspectives have been suggested, but the different schools of thought have yet to agree on the correct one.
Given a group of randomly chosen people, how big is the chance that two persons in this group share the same birthday? You can assume that each day of the year is equally probable for a birthday.
The answer might surprise you. In case of a group size of 23 people, the chance that there are two people in there with the same birthday, is already 50/50. A probability of 99% is reached with a group of only 57 people!
Imagine I had a very long piece of string: long enough to wrap it around the equator of the Earth. And I'll do just that. Yikes — that's about 40,000 kilometers (or 25,000 miles) of string! I will make sure its pulled completely tight and connect both ends to each other: the result is that it lies flat onto the surface.
Now let me extend the string with just one measly meter. Compared to its total length that's not a lot, is it? Once again I pull it tight, but now, with the added meter, it has come off the ground just a tiny bit. Assume that this extra distance (between the string and the Earth's surface) is equally divided and thus the same all around the globe. How much would you guess this distance is? Surely this must be in the order of nanometers or whatnot, right?
Well, no. Turns out, the entire string has now come almost 16 centimeters off the ground.
