Apples

[shinny]

Hahaha I'm staying up for /0. He's been at it for over 2hours now on "who's online" He shows promise

Jokes on you if he just afk'd LOL. I'm expecting some good quality posts when I come back tmrw though...this is starting to get interesting =T

[TrueTears]

If /0 has been typing constantly for 2 hours, his post is gonna be massive. Probs some uber pr0 way of solving this Q.  8-)

....or.... he got bored and went straight to his happy time.

or maybe he took a flight back to kazak to visit the PM.

[kamil9876]

lol yes I was thinking the same thing, I'll use this spare time to improve my stalking skillz so that I can see if he really is working on it.

edit: I was refering to shinny's post, I dont want to know what goes on in "happy time" D:

[/0]

Yeah kinda massive post with inequalities but then i just cbf... I still have no idea how you got 750

[TrueTears]

Yeah kinda massive post with inequalities but then i just cbf... I still have no idea how you got 750

Ohhh at least post it, 2 hours of posting and you delete it all, that's a waste

P.S you know how much kamil was waiting for your post? He feels quite disappointed now.  :knuppel2: :knuppel2:

[/0]

hmm I thought of using 2 checkpoints.
On the first journey you set up a checkpoint at .
On th second journey you some of the apples from this checkpoint to setup another one at .

Somehow I got , but that doesn't make sense or help me because so I needed a lower limit on a and b.

[shinny]

Sigh, random burst of inspiration whilst attempting to try to get to sleep. Haven't proof read this working out, but it seems roughly about right I hope.

It's basically what I did before; with 1000 998 checkpoints. The aim is to ALWAYS be carrying 1000 apples within the truck where ever possible. And now to hopefully get a good night's rest now that this is off my back. Also, I assume (or maybe it's the sleep deprivation talking) that this can be improved upon when you don't use integers for the number of checkpoints (e.g. a checkpoint every half km, or with an infinite number of checkpoints even), but I'm not going to bother thinking into it too much.

EDIT:
OK since SMF isn't letting me post for some weird reason, I'll just edit this in and expand on what I had last night and hopefully prove it (to myself even, I might have made a mistake)...

My working out last night had 2 distinct phases.
Phase 1: 0-500
You've got 3000 apples at A, and we're moving 1000 apples at a time, 1 kilometer at a time. If you move 1000 1 kilometer away, you've got 2000 at A, and 999 1 km away. Repeat this step to get 1000 at A and 1998 at 1 km away. Now move 1000 of the apples at the 1 km mark to the 2 km mark, so you've got 1000 at A, 998 at 1 km and 999 at 2 km. Now of course, move the 1000 at A to the 1 km mark, and you're at 1997 at the 1 km mark and 999 at the 2km mark. This of course follows with 997 at the 1 km mark and 1998 at the 2 km. If you keep repeating this 'shuffle' ,you'll notice that at the step I left off at, the number for the lower numbered checkpoint decreases by 2, but the greater number checkpoint stays constant at 1998 (look at rows 6, 9 and 13). Using this pattern, if 1 is 997, 2 is 995, 3 is 993, then basically if x is y..., which means at x=499, y=1; which is the step I skipped to.

Phase 2: 501-1000
Ok, screw the 1 apple, just ditch it. If we go back to fetch it, it'll be eaten anyway. Time to progress with the 1998 at 500 km. Do the same shuffle movement, just easier; pick up 1000 apples to the 501 km mark to get 999 there, and 998 left at the 500 km mark. Move the ones left behind at the 500 km mark, and we get 1996 at 501 km. Obviously, we're losing 2 apples per kilometer travelled (which makes sense, since we're making 2 distinct trips whilst carrying apples for each km). So if 500=1998, 501=1996, then using x=y again..., so at x=998, y=2. This means the final few steps of this process would be;

So, 999 apples. Anyone see anything wrong with this working out? Otherwise I guess I'm now in the lead =T

[/0]

I did some research and apparently 533 is the best the source can come up with http://www.keymath.com/documents/daa1/CondensedLessonPlans/DAA_CLP_00.pdf

[TrueTears]

I did some research and apparently 533 is the best the source can come up with http://www.keymath.com/documents/daa1/CondensedLessonPlans/DAA_CLP_00.pdf

Nice document, is that part of a book or something?

[/0]

I did some research and apparently 533 is the best the source can come up with http://www.keymath.com/documents/daa1/CondensedLessonPlans/DAA_CLP_00.pdf

Nice document, is that part of a book or something?

dunno random googling lol

[kamil9876]

Okay then.... here it is:

Take 1000 apples and put them at 500km mark. (we have 500 apples at 500km). Now take the next 1000 and put it at 750km mark(250 at 750km now). Now take the final 1000 apples and go to 500km. Pick up 500 apples (so you are full now). Then go to 750 and pick up the 250(you are full now). Then you just have 250 to go and hence 750 will be dropped off at B.

Notice how this is an extension of 2000apples at A. Because it's the same things just no 250apples at 750.

A lot of ideas sparked from this, but just like you I had a lot of inequalities and equations etc. But I think I have made a few interesting lemmas that are on the way to proving the general algorithm for any initial condition at A.  

[evaporade]

Sigh, random burst of inspiration whilst attempting to try to get to sleep. Haven't proof read this working out, but it seems roughly about right I hope.
(Image removed from quote.)
It's basically what I did before; with 1000 998 checkpoints. The aim is to ALWAYS be carrying 1000 apples within the truck where ever possible. And now to hopefully get a good night's rest now that this is off my back. Also, I assume (or maybe it's the sleep deprivation talking) that this can be improved upon when you don't use integers for the number of checkpoints (e.g. a checkpoint every half km, or with an infinite number of checkpoints even), but I'm not going to bother thinking into it too much.

EDIT:
OK since SMF isn't letting me post for some weird reason, I'll just edit this in and expand on what I had last night and hopefully prove it (to myself even, I might have made a mistake)...

My working out last night had 2 distinct phases.
Phase 1: 0-500
You've got 3000 apples at A, and we're moving 1000 apples at a time, 1 kilometer at a time. If you move 1000 1 kilometer away, you've got 2000 at A, and 999 1 km away. Repeat this step to get 1000 at A and 1998 at 1 km away. Now move 1000 of the apples at the 1 km mark to the 2 km mark, so you've got 1000 at A, 998 at 1 km and 999 at 2 km. Now of course, move the 1000 at A to the 1 km mark, and you're at 1997 at the 1 km mark and 999 at the 2km mark. This of course follows with 997 at the 1 km mark and 1998 at the 2 km. If you keep repeating this 'shuffle' ,you'll notice that at the step I left off at, the number for the lower numbered checkpoint decreases by 2, but the greater number checkpoint stays constant at 1998 (look at rows 6, 9 and 13). Using this pattern, if 1 is 997, 2 is 995, 3 is 993, then basically if x is y..., which means at x=499, y=1; which is the step I skipped to.

Phase 2: 501-1000
Ok, screw the 1 apple, just ditch it. If we go back to fetch it, it'll be eaten anyway. Time to progress with the 1998 at 500 km. Do the same shuffle movement, just easier; pick up 1000 apples to the 501 km mark to get 999 there, and 998 left at the 500 km mark. Move the ones left behind at the 500 km mark, and we get 1996 at 501 km. Obviously, we're losing 2 apples per kilometer travelled (which makes sense, since we're making 2 distinct trips whilst carrying apples for each km). So if 500=1998, 501=1996, then using x=y again..., so at x=998, y=2. This means the final few steps of this process would be;
(Image removed from quote.)
So, 999 apples. Anyone see anything wrong with this working out? Otherwise I guess I'm now in the lead =T

in the first table, From step 9 onwards all wrong

[evaporade]

My answers

(1) 834

(2) 3666

Another question: If the capacity of the truck is 999, what is the max delivery to B?

[kamil9876]

Can you just tell me the steps the truckie must go through? Don't worry, I'm not asking for working out, the steps are the answer. ie: consider the question as "what strategy yields maximum" rather than "what is the maximum"

[kamil9876]

Ahh yes evaporade is correct about shinny's mistake, truckie forgot to eat one apple at each step of your iteration. ie: in step 9 you did: 997+998 whereas really it shouldve been 996+998 because he wouldve eaten one apple when moving his load 1km. You made this same mistake at step 13 but I havnt analysed your method any further yet. I will have to think more hehe