The Popcorn Theorem

This experiment started about a month ago, when I noticed that no matter at whichever setting you pop the microwave popcorn, there are always some kernels left unpopped. So either there must be an optimal setting of the microwave OR these kernels should not be there in the packet.

I tried to pop various packets at different power levels at different times in the Microwave. The recommended settings are usually printed on the back.

First we have to come up with an estimate of the number of kernels in each packet.

M = avg. mass of 1 kernel = 0.17294117647 g

M_p = mass of the whole packet = 100g

M_s = Secondary mass of [palm oil, edible vegetable oil, iodised salt)

tomato chilli seasoning (tomato powder, spices, iodised salt, sugar, yeast extract powder, hydrolyzed vegetable protein, binder (dextrin), acidity regulator (citric acid), flavour enhancers (disodium guanylate, disodium inosinate), onion powder, garlic powder, anti-caking agent (silicon dioxide), added flavour, nature identical flavouring substances)] + [Packaging]

So now M_p = M*(number of kernels) + M_s

On an average the mass of all kernels is known to be about 85% of the whole packet = 85g.

This gives us:

Average number of kernels = 491.49 ≈ 491 kernels for all practical purposes in each packet.

And then I counted the number of unpopped kernels in each case.

However, before we proceed we have to define a variable that takes into account the number of burnt popcorn as well. This is because just to reduce the number of unpopped popcorn we can’t increase the temperature so much that the popcorn burns – and is therefore inedible. Total number is taken as 491 in all cases.

N_p = Number of popcorn that are properly popped and palatable :p

N_u = Number of unpopped kernels

N_b = Number of burnt popcorn in the final result

Therefore popping efficiency η_p (%) = N_p/491 = (491 – N_u – N_b)/491

So we can combine the number of unpopped and burnt popcorn into just one metric – η_p

Here is the data:

So to minimise waste we see that if a popcorn packet is popped at

650W then the ideal number of popcorn kernels should be 491-113 = 378 kernels out of which 7 will burn and the consumer will get to eat 371 popcorn.

800W then the ideal number of popcorn kernels should be 491 – 44 = 447 kernels, all edible.

1000W then the ideal number of popcorn kernels should be 491 – 97 = 394 kernels, all edible.

That is OK but the world is less than Utopian and so we must make allowance for someone who leaves the popcorn in the microwave for some extra time or simply just does not follow temperature instructions, does not know his/her microwave settings or whatever. Can we arrive at an optimum number of popcorn kernels to put in the bag?

Efficiency is just a measure of least wastage and does not mean that you get to eat maximum popcorn!

The most number of edible popcorn (>400) can be obtained are only at 800W when heated between 2:40 and 3:20 only.

So though the popcorn manufacturer would like to go for maximum efficiency the consumer would like to go for most popcorn eaten. This does not need to be a Principal-Agent problem as we already have a solution.

And the solution is:

The popcorn company should reduce the number of kernels from 491 to at most 450. There is no power setting and no time at which more than 450 popcorn will be edible anyway!

A saving of 41 popcorn per packet = saving of 41* 0.17294117647 = 7.09058823527 gms per packet.

Considering that 1 tonne of corn kernels cost about $1800.

Americans consume about 17 Billion quarts of popcorn each year. That’s 13.2 Billion kgs = 13.4 Million Tonnes.

Considering that there is a mass of 491*0.17294117647 gms in one packet, 13.4 Million Tonnes is about 157 Billion sachets of popcorn.

In a nutshell, just by reducing the number of popcorn by 41 in each sachet or the mass of popcorn by 7 gms per sachet, we have 78 gms of popcorn per sachet (instead of the usual 85 gms). A saving of 8.235% which is 12 Billion popcorn sachets EXTRA at no cost at all.

OR

That’s a collective saving of over  $2 Billion for all popcorn making companies! :)

And the consumer – don’t worry, you’ll get your 400 odd popcorn in every sachet if you just keep to 800W between 2:40 and 3:20 :)

And all you consultants out there – don’t you dare copy this research and sell it to ACT-II for a couple of million dollars. Read the disclaimer first.