BMAD

Let the length of a domino be 1. On the edge of a table, a single domino can extend out a distance of 1/2 before being unbalanced. A stack of two dominoes can extend out a distance of 3/4. How large of an overhang can you create with 5? 10? N? How many dominoes are needed before the top domino is one full horizontal length away from the edge of the table, for a full overhang length of 2? Each domino must be placed in the same orientation (don't rotate the dominoes or stand them on end.) this can also be done with playing cards

for B, of pairs of whole numbers, what is the smallest m+n where (m,n) and (p,q) have the same lcd and gcf?

yes. any type of real number (no complex)

Classic magic number square problem but hopefully with a slight twist. If all the rows, columns, and diagonals add up to 33 using any number you desire. Is the center uniquely determined?

A while ago, I gave my Pre-algebra class a puzzle problem: "Given two natural numbers, m and n. If their GCD is G=6 and their LCM is L=72, what are the numbers?" a) What were all possible (m, n) for G=6 and L=72 (m<=n)? b) What's the smallest sum, m+n, for any (m, n) pair that share the same G=gcd and L=lcm (with another m<=n; G>=2)? c) Find the (G, L) pair with the most solutions (m, n) for the same G=gcd(m,n) and L=lcm(m,n) (G>1, L < 1001).

My friend's music fan website started out fast: lots of visitors the first week. But every week, there were fewer and fewer; the second week there was one more visitor than 2/3 of the first week, the 3rd week there were 2 more than 2/3 of the second week, the 4th week there were 3 more than 2/3 of the 3rd week, and so on, until the final week, when there were just 533. How many weeks was the website up? How many visitors did it get the 1st week? How many total visitors were there?

Brilliant observation. So does the number of worst case scendarios increase in a pattern? Does this method tell us that each worst case is equivalently bad regardless of algorithm?

For rules on self refrential numbers see: As per a special request, what is the largest sellf refrential number? And yes there actually is such a number.

Greedy (as many mate in a single instance as possible). Blood relatives means that they have a genetic relation.

The biggest white area or all of the white areas. The sum, I am almost certain they approach a limit. By the way if anyone was wondering about this problems category. It belongs to set of mathematics known as fractals. Specifically iterated fractals of self similars.

Six unrelated beings randomly selects their mates (three female and three male). Where 1 female mates with 1 male. The result of such mating produces two offspring, one male and one female from each mating. Once the new generation comes of age, the random mating event happens again except blood relatives cannot mate. the process repeats. The beings die upon surving through the production of the second group of offspring. How many generations until the creatures can no longer mate? If they can always choose a partner, why? Assume that all creatures will choose a mating situation, if one exists, where everyone mates.

Bear in mind that I don't mean a straight line necessarily. Just a connected set of line segments

The knobs won't turn beyond the boundaries

I have two points on a cartesian plane with bounds from (-8,8) for x and y. One point is at (-7,2). I drew a line from this point to another point using etch-a-sketch moves (left/right) and (up/down). Each turn of a dial is a one unit move. I moved the dials a total of nine times. What are alll the possible endpoints?

Clarification on the op: We are assuming that the user has knowledge of every possible sorting algorithm at hand. Armed with this knowledge there will still be a minimum number of 'steps' needed to fix the dvds. We seek to find this minimum number assuming that there is a universal worse case configurationn of dvds. Universal in the sense that if an algorithm is applied to this array it would take more moves to fix (or equal to) than any other array.

Deism allows your interpretation as well. Ps. My point on the last part was just to show the counter argument towards a perfect god

In terms of practicallity, they both provide satisfactory results. I think it would depend on the arrangement of the dvds. In your case, both produced two moves so it doesn't matter but I am unsure if that will be the case in larger samples and in all arrangements. Is there a way to set it up where both strategies run and the one with the better score be kept?

Umm...what? I mean, if there is a repeatable pattern that you follow in order to solve a problem, which in the case of the OP there is, then you can write a computer program to solve it (that's what programs are...repeatable steps) From my understanding of the OP (and forgive me, because I may have missed something in the intermediate posts), the way this sort works is basically you grab a DVD, you KNOW what position it should be in, and you insert it there. So for example: Define this as a single "move": Look at a DVD...does it belong at the beginning of the list of DVDs? If not, take it out...shift all of the DVDs between its old position and its new position over and insert it at its correct spot. Now the OP also says they don't care about ascending or descending order...whichever takes fewer of the above moves (but WORST case...in other words, take the WORST POSSIBLE permutation of the DVDs using the above sort, and figure out the fewest number of "moves" it would take to sort them either ascending or descending...that is the REQUIRED number of moves to ENSURE you have sorted them.) This is interesting. It seems to support the n/2-1 conjecture mentioned earlier. Is there a way to see the solution (or shifts made) to the 11 dvd problem organized like gavinksong did?

Gavinnksong Do you agree with these notions for this problem: 1. Even sets have a max of n/2 moves to sort given the wose possible disarray of DVDs. 2. Odd sets either have n/2 +/- 1 moves to sort given the wose possible dissarray 3. WLOG, The worst possible disarray apears the reversing of dual-consecutive number where if there are k terms and k is odd the kth term goes in the middle. Example of #3: 1-K terms where k is even 2,1 , 4,3 , 6,5 , ... , k, k-1 1-k terms where k is odd 2,1 , 4,3 , 6,,5 , ...n, n-1, k, n+2, n+1, ... k-1, k-2

Not quite. God intentionally or unintentionally created our universe's existence. To bring order to the chaos, as seen in many faiths there seems to be a drive as such, they created a set of rules (or perhaps the rules exist as a result of how the universe was created). That is the extent for which God directly interacts with Earth. Think of the Christian creation myth but instead of looking at it applied to earth and humans, apply it to a grander scale--the creation of the universe and sentience. If we take the Christian belief as God being an all-knowing, all-caring, all-powerful, all-present god then god would have known from the start everything that needed to be done to care for "his" humans and would have made sure that everything was in place (the master plan) at time of creation. If this is true, then what is the purpose of there presence? It can't be to watch someone get murdered (for that would contradict all-caring, it can't be to observe us make a decision and judge us, as that would contradict all-knowing since he already knows and therefore judged us, it can't be to ensure his laws are followed for that will contradict his all-powerful status as that would imply we superseded god's law. Therefore, including the characteristic of all-present in that matrix of how god 'is' produces a string of contradictions and hence, deist reject that defining aspect of god. Not saying that God is never present but always present in the minutiae of everything is unnecessary and illogical.

So are you saying that there isn't a best 'worst' disarray that regardless of algorithm will produce more moves than any other such disarray when comparing effect of said algorithm on both disarrays?