Thursday, June 21, 2012

The Ketchup Problem

MIT scientists are working on a bottle of ketchup where you CAN get out every last drop.  See here for details and some nice videos of the new bottle in action. This will be used on other products (like Mayonnaise) and will save much thrown away food every year! However, IF this had been around 45 years ago, then I might never have gone into Math!!! Why is that?

When I was 8 years old I had the following thought:

When ketchup is at the end of the bottle it drips very slowly.  It never stops dripping hence the number of drops is infinite.  Could I put an empty bottle of ketchup below it and catch all of those drops and hence never run out of ketchup?

When I was in college I came up with the following explanations:

1. NO: By the same reasoning there will never be an empty bottle of ketchup!
2. NO: Each drop is half the size of the previous one so even though this is an infinite series it converges.
3. NO: The time between drops doubles so it would take infinite time to fill up another bottle.

To this day I will NOT give up my conviction that the number of drops is infinite.  This ketchup problem may be the first time I saw a real world phenomena and made a math problem out of it. I blogged about other such problems here and here.
The ketchup problem so intrigued me that I became a math major. OR the fact that I thought of this problem indicates that I liked math and hence went into it. I would say its a chicken-and-egg thing but we need a new metaphor since
the chicken came first.

Other questions I thought of as a kid:

1. Why do people smoke tobacco if its so well known that its bad for them?  This one still puzzles me.
2. If, as the TV ads indicate, people want Margarine that tastes JUST LIKE BUTTER then why not just get butter? I NOW know that Margarine was cheaper back then (still is).
3. If, as the TV ads indicate, people want instant coffee that tastes JUST LIKE GROUND ROAST, why not just get ground roast? I had no idea what any of those terms meant as a kid. Why I cared is a mystery since I didn't drink coffee as an 8 year old (and I still don't).  I now deal with math terms i don't understand. Why I care is a mystery.
4. How is eating everything on my plate going to help the kids that are starving in Africa? Mom never quite answered that one.
5. How come whenever I see six kids playing together there are always either three that get along or three that don't get along?

Had I pursued the first three questions I would either be in sociology or advertising. Had I pursued the fourth question I might be in economics or politics. I have no idea what pursuing the fifth question would have lead to. Child Psychology? Party planning?

READERS- do you recall a math problem that you cared about as a kid that likely either inspired you to do math or indicated that you were interested in math? Note that you don't need to have SOLVED the problem as a kid- I am interested in your interests,
not your ability (note that I didn't any of my problem as a kid).

1. I got obsessed as a kid by the question of the best strategy for attacking another country in the board game Risk. This involved rolling either one, two, or three dice, and your opponent rolling either one or two, with the constraint that you had to have that many "armies" in the country. One then compared the two highest, and next two highest, dice to see who lost an army. I was pretty sure the best strategy was always to roll as many dice as possible, but wanted to know for sure. This led me to try to extensive fooling around with discrete probability, game trees, etc.

2. Well, the 5th question is related to a problem in graph theory, isn't it? It is easy to show that for every graph with 6 vertices, there is a triangle either in the graph or in its complement. So perhaps you could have pursued it and still come out as a mathematician :)

3. Speaking of Ketchup, may I give you the link of an excellent article http://iml.univ-mrs.fr/~girard/mustard/article.html
" Mustard watches : an integrated approach to time and food"
It also speak a little bit about ketchup

4. As a teenager I wondered how far away the horizon is, and realized that this could be computed from the dimensions of the earth (modulo refractive properties of the atmosphere). With some encouragement from my grandfather (and use of his computer to look up the dimensions), I calculated that the horizon is about 5 miles away for a 6-ft tall person standing at sea level.

5. As a kid for a very long time I was obsessed with perpetual motion machines. I had pretty much convinced myself that the rotor in an electrical generator would continue to spin by itself producing electricity for free if only internal friction could be circumvented. Only much later when I was finally in engineering school I finally understood the concept of magnetic field induced by the electricity that gets generated which in turn produces the primary component of 'drag' on the rotor. A principle that is so very obvious that even today I feel embarrassed about my naivety in earlier days. I do often wonder, however, what it is that keeps creationists stuck to their kool-aid, smokers to their tobacco and a significant number of car owners to their gas guzzlers.

6. @GASARCH I am at the least disappointed about ur blog post quality, it is deterioriating. If u wanted to report something interesting then why don't do something more relevant.

7. Surely you don't think today that there are an infinite number of drops of ketchup. Ketchup is only ketchup as long as the drops are at least big enough to contain molecules of the stuff. Beyond that, your "drops" are atoms of hydrogen, oxygen, tomatogen, etc, and beyond that you're just putting protons and neutrons on your burger.

1. There are molecules
Then protons/neutrons/electrons
Then quarks
Then- who knows what those physicists will tell us next!

8. I remember being constantly befuddled by problems which I knew could be modeled mathematically but I honestly didn't have the analytic tools I have now, mostly game theoretic problems. i.e., as a kid I always wished I could figure out some super-efficient rock-paper-scissors algorithm which would make me unbeatable (and thus obviously the coolest kid in school). Alas, I now know that simple randomization is the most effective process.

I also distinctly remember trying to generalize arithmetic functions as a high-schooler: I figured, if addition is just repeated succession and multiplication is just repeated addition and so on, then couldn't we just have a general function like f(a,b,c) such that if c=0 then a and b are added, if c=1 then a and b are multiplied, etc.? This literally kept me up all night, as I found myself needing more and more general functions to truly generalize. I never ended up coming to a conclusion that satisfied me (in a really similar way to how Cantor's transfinite ordinal and cardinal numbers always failed to satisfy me, I'm not entirely sure why...), though I remember that curiosity being rekindled while learning about the "arithmetical hierarchy" in your Theory of Computation class. Come to think of it, if you or anyone else have/has recommendations for readings which may shed more light on this topic please let me know!

9. Surface tension means that the drops effectively have a minimum size, which is why you don't see ketchup dripping out molecule-by-molecule, let alone quark-by-quark.

10. 2nd and fourth questions I used to think also.. The question you started with came to mind also.. but I don't know how a child gets their intuitions in mathematics but I had some feelings that it can take infinite ('too much' for a child) time.

I also had some foolish concepts which made me apart from chemistry.. I used to think how it is possible to write 'Ca' over an Atom of Ca.

I also used to think that mathematics can describe everything, so when I couldn't get answers from my teacher I used to think he doesn't know it.

I can't remmeber.. but I am sure there should be some interesting thoughts..

11. When I was a kid growing up in Oklahoma City, my mom told me that at 12 noon, the sun was exactly overhead. Since 12 noon in Oklahoma City was also 12 noon in Tulsa, I didn't understand how the sun could be directly overhead both Oklahoma City and Tulsa at the same time.

12. When I was in middle school a problem was left on the board which read, when is 6*9 = 42. I had been exposed to binary at some point and eventually found that in base13 6*9 would equal 42.