BMAD

How many random shuffles does it take to randomize a deck of cards?

Suppose n is an integer in base 10. Note that n2 ends with 0 if and only if n ends with 0. Now consider integers represented in base b, where 5 <= b <= 9. Determine for which base b (if any) the following statement is true: For any integer n, n2 ends with 0 if and only if n ends with 0.

Along the edge of a perfectly circular piece of wood having uniform density we choose N 3 random points at which we attach slender legs of negligible mass to form a table. What is the probability that this table will stand without falling?

This one may have been posted before: why are some letters on top of the line and others below the line?

brawn

oops I accidentally included the answer in my question . Often when people think of this problem without actually trying it they believe that the coin only rotates halfway or 3/4's of the way and don't realize that the coin makes a complete rotation. My mistake guys.

What about the other part of the conjecture that the area is half?

Excellent idea! Now carry it out.

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.

care to explain your approach?

oops. nice catch. I forgot to upload it. but it is meant to just show an arbitrary quadrilateral, see below:

day 8

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?

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?

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?

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.)

Drawn

A couple weeks ago, I created a question requiring the shortest path. Now for this question, assume that roads can be curved. We need a road that can pass through the following four cities (location of each city listed as coordinates): Los Angeles (3,4), Newport Beach (5,1), Pasadena (4,5), Santa Monica (2,3). a) What's the smallest degree polynomial y=f(x) that will pass through all four cities? . . . b) What is the exact equation of this polynomial? (Hint: use fractions not decimals) c) Would this road go through Chatsworth at (1,6)?

care to explain your answer?

My answer is most like yours but my sector areas were different (for the 4cm and 5cm circles).

I think someone has an issue with me. I have been working hard to keep the forum questions alive but someone or maybe several people have gone through and have marked down every entry i have posted. This upsets me and makes me not want to continue participating here.

but are these the only ones that work?

for some reason i am getting 20 1's up to 100 (not counting 100). I will recheck my numbers: 1-9 =1 10-19=11 20-99=8 I still get 20.