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My grocery store frequently has a special on pears and apples where you fill the bag for a set price and I'm never quite sure how well I've done maximizing the contents.

My restrictions and (limited) sense of etiquette:

  1. I prefer apples to pears, so I wouldn't want a solution of more than 1/3 pears.
  2. I also get bored with one fruit, so I don't want 0 pears.
  3. I also refuse to harm any fruit in the process (i.e. no apple sauce.)
  4. The bag is 2 sheets of 30x30cm plastic when laid flat. Apples are 4-6cm ~spheres, while a pear might be 4x6cm.
  5. I have no measuring instruments in the shop, no willingness to replace fruit already in the bag, or take an excessive amount of time, etc.

Given these restrictions, my question is how do I maximize the total volume filled and minimize the air gaps assuming these varying sphere and pear shapes?

Do I start with smaller or larger apples first, do I start with pears first? Are same size or mixed size/shaped objects easier to use to fill a space? Etc.

More about the bag:

It has plastic handles at the top which need to be closed together making a rough sphere. It is prone to ripping, particularly where the handles connect, so stretching is not practical. The bottom corners are pushed up inside about 2 cm and welded in the side welds, making it slightly more spherical when full. The offer claims that it can hold at least 3 Kg.

Bag with vegetables

  • How rigid is the bag - what shape does the bag take when loaded? Is it rectangular in cross section, or does it become a distorted sphere? – John Feltz Oct 29 '16 at 1:40
  • @JohnFeltz I've added a better description of the bag and an example of it holding a similar vegetable offer. – lossleader Oct 29 '16 at 10:42
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Considering that in a bag of those dimensions & taking the individual objects' sizes into consideration, that there will only probably be one piece of fruit difference in an optimal vs non-optimal filling, & the fact that you don't have time/inclination to keep replacing items until fully satisfied, you have to my mind only two potential methods...

  1. Choose smaller fruit. The gaps will be smaller, more randomness is possible & pure luck could find the right gap in the bag to get that one last piece in.

  2. Like settling a box containing flour or cereal, which is much easier to visualise, the item size being much much smaller & the settling therefore more obvious - shake it or bang it [gently] against a horizontal surface.
    This will give the items a rough chance to naturally find their own best levels.

... a late 3... do 1. a few times on separate visits with clearly differing sizes. some same, large or small, some mixed weights & see which comes out heaviest when you get home.

To really be sure you're onto a winner, you'd have to weigh the cores afterwards too, & subtract from the arrival weight.
No good if you can get 3kg in the bag, only to find later that 2.5k of "better" sizes would produce less waste overall

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Choosing all large or all small fruit does not change the packing density. But if you choose some large fruit, with some smaller pieces that fit in the interstices amongst the larger, you can increase the density. See Wikipedia on sphere packing.

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