BMAD

Assuming the individual is making the best moves possible and the DVDs are in the worst possible disarray, how many moves will it take to put the in alphabetical order?

I find this question ironically intriguing as I was honestly just sorting 22 DVDs on two shelves and wondered, as I hope I am not the only nerd who does this, "is there an efficient algorithm for doing this?"

If it were to change it would have to change at an odd like 7 since 6 being even would result in n/2 changes, correct?

Great point. My program allows me to tweak the ratio (or pull) to make the 'hard to see' patterns in the above drawings easier to see. I assure you the image is there yours just has more noise. But if you examined frequency by location (or clustering) the images wouuld clean up like mine. But you are right, the empty spaces in my plate examples aren't completely empty.

I am neither. I am a deist. I do not deny the existence of god. I question the active role they play.

So the only point that could ever end up in the center of the shape is the initial point but the resulting image would be the same as no other dot could go in that empty space.

Look backwards. If the second point were in the middle empty area, where does that put the first point? Maybe this logic can be extended to the other smaller empty areas as well... I can't look backwards from the initial point. If the initial point were in the central area, the second point would not also be there. I asked you to look backwards from the second point. if the second point is in the center, then there could not have been an initial point as any where twice the distance from the center is outside of the shape, therefore we could not under the rules and point system have points in the center.

I never said you were wrong about rounding up. I was just pointing out an apparent contradiction in my chart and your answer. Perhaps Dvds 1-4 is trivial and the round up principle starts at five,,, checking n=5, ABCDE scrambled as BAEDC AEDCB EDCBA (two moves) scrambled as BEADC EADCB EDCBA (two moves) scrambled as BADEC ABDEC ABCDE (two moves) scrambled as ABEDC ABCED ABCDE (two moves) now i did not exhaust all of the possible arrangements of five nor do i claim that i found the worst case scenario but it does appear to only take two moves. So is there a point where it switches from n/2 (round down if decimal) to n/2 (round up if decimal) or is eleven purely unique? and if it is unique, why is it unique?

octogons anyone?

Here is the effect of applying the 'game rules' to the pentagon in the shape of a home plate. See that the empty space pattern in the pattern of the starting figure occurs again. Also notice that this shape is irregular and not triangular.

Does this patterning of similar shapes occur in other figures as well (like a square, rectangle, octogon, etc.)? Or is it relegated to only triangles?

Look backwards. If the second point were in the middle empty area, where does that put the first point? Maybe this logic can be extended to the other smaller empty areas as well...

What is the smallest (by value) self referential number? Obviously it can't be 0 or 1. See for the rules of self-referential numbers.

Nicely done Anza! For other people to see the "pattern" forming, you need to let the above html run for about five minutes to clearly see it. The longer it runs, the better.

Don't erase your prior points, show them all as each new point is found

This is an excellent program for my programmers out there but anyone can successfully examine this: The Chaos Game: 1. Label an isosceles triangles vertices 1,2, and 3. 2. Select some form of a randomizer that will pick the numbers 1,2, and 3. 3. Pick a point inside the triangle and put a dot there. 4. Use your randomizer to select one of the numbers from 1,2, and 3. 5. Place a dot midway between the vertex with that number and the current dot. 6. Now, using that new dot as a reference point, repeat steps 4, and 5. 7. Continue these trials until you notice something magical (hopefully) Why is this happening?

Are you sure we are rounding up? and of course we are ignoring the trivial cases of 1 dvd and 2 dvds.

this is one approach

Excellent...continue

Consider completing the table, maybe a pattern will form: DVDs moves 1-0 2-0 3-1 4-2