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Show that if the difference of the cubes of two consecutive integers is the square of an integer, then this integer is the sum of the squares of two consecutive integers. (The smallest non-trivial example is: 83 − 73 = 169. This is the square of an integer, namely 13, which can be expressed as 22 + 32.)

Start with any general quadrilateral, and connect the midpoints of consecutive sides, making an inscribed quadrilateral as in the diagram. That inscribed quadrilateral, in the diagram, seems to be a parallelogram. Let me conjecture that this inscribed quadrilateral is a parallelogram with half the area of the original quadrilateral. Can you prove or disprove either part of my conjecture?

We have two identical coins. And we roll the one on the left halfway around the other coin, so it rotates without slipping against the other coin, so that it ends up on the right of the other coin. It has rolled over a length of only half its circumference, and yet it has made one complete rotation. Which way is the head of the coin facing?

480608 , 508811 , 723217. These three numbers, when divided by a certain natural number > 1 , all yield the same remainder. What is that divisor and that remainder? The 'best answer' will be awarded to the person who can develop an elegant method that does not utilize brute force or code.

Recently a self-proclaimed stand-up logician held a show with thirty people. here is his opening dialogue: Hello folks. It's an honor to be on Amateur Logician Night, here on the Internet. How many of you are from out-of-town? Let's see a show of hands. Now, the people who raised their hands may be truth-tellers who are from out-of-town, or you may be liars who are from in-town. As the amateur logician, it is my job to determine who is really from out-of-town, and who is not. This is not an easy job, as you can well imagine. If I were to ask how many of you are truth-tellers, then all of you would raise your hands. The liars would have to lie, and claim to be truth-tellers. Now, let's see a show of hands, all of the people who raised your hands the first time, when I asked how many of you were from out-of-town. Very good. All of you, who now have your hands up, are from out-of-town. Some of you are truth-tellers who raised your hands both times. You are from out-of-town. Some of you are liars who did not raise your hands last time. You too are from out-of-town. So, all of you who now have your hands up, are from out-of-town. Thank you. Thank you. What a crowd. Thank you. Would this approach work in identifying those that always lie and those that always tell the truth? Why or why not?

[spoiler='Slight change in the process does it']Have them stand in a straight line. Instead of requesting a show of hands, ask them to take a step forward. Liars will step forward at one or the other request. Truth-tellers from out of town will take two steps. Truth-telling locals wont' move. The one-steppers comprise all of the liars.

No need to spoiler. If it has made a complete revolution, (and it has) it's facing the same "way" after as before. Maybe I'm missing something.

Which diagram? I'm not seeing one.

common factor of difference between pairs of numbers