Tag Archives: Dumb crap

A Math Degree in the Kitchen

So today I was making some rice, and I decided based on calorie load that I wanted more than 1/3 cup of rice, but less than 1/2.  I decided that 5/12 cup was acceptable, but my only tools were a 1/4 cup scoop and a 1/3 cup scoop.

I didn’t have to think very hard about this; I instantly knew it was possible to get 5/12 cup of rice with these tools, because 4 and 3 are relatively prime–i.e., they share no common factors.  More on this in a minute, but first, how to do it.

It’s a very uncomplicated idea, and it should look very similar to the Diehard 3 water puzzle.  In the film, John McClain has a 3 liter jug and a 5 liter jug, and wants to get exactly 4 liters of water.  My rice puzzle is basically the same game.    To get my 5/12 cup of rice, I fill the 1/3 cup scoop and pour that rice into my rice cooker.  Then I again fill the 1/3 scoop and pour that (without spilling) into the 1/4 scoop.  How much remains in the 1/3 scoop?  Well, I had 4/12 cup of rice in it, and I took out 3/12 cup of rice.  So only 1 remains in the 1/3 cup scoop.  Add the 1/12 to the rice cooker, and since 1/3+1/12=5/12, we’re done.  And although that’s how I did it, I actually didn’t even need to get the rice cooker involved.  I can form 5/12 cups inside the scoops without having to use another vessel (though the thing holding all my rice is allowed).  I could have filled the 1/3 cup scoop, poured that into the 1/4 cup scoop, emptied the 1/4 cup scoop, poured the 1/12 cup in the 1/3 cup scoop into the 1/4 cup scoop, then filled the 1/3 cup scoop.  Tada!

I warned you that it was very uncomplicated; the above is really not very interesting on any level.  What is interesting is why this works, and how I knew so quickly that it must be possible to do it.  If you don’t know why, then all that stuff I just wrote might seem like a cute curiosity, when really there is a very basic underlying principle involved.  Instead of needing to ask “I wonder if it’s possible to get 1/6 cups of rice with those tools”, I can instantly say that the answer is “yes”.  I didn’t have to think about it at all.  Here’s how it works.

Instead of thinking of the scoops as 1/4 and 1/3 cup, I’ll do you a favor and eliminate the fractions for you.  Let’s think of them as 3 and 4 serving scoops (out of 12), respectively.  Believe it or not, we’re almost done.  As it turns out, if the greatest common divisor (written gcd) of these two values (in this case 3 and 4 from 1/4 cup and 1/3 cup scoops, respectively) is 1, then it’s possible to get any of 1/12, 2/12, 3/12, 4/12, 5/12, 6/12, or 7/12 cups of rice (maxing out at 3+4=7, because that’s the maximum amount of stuff our two scoops can hold).  Said another way, I can get 1, 2, 3, 4, 5, 6, or 7 servings (out of 12) in this fashion.  Why?  Because if gcd(3,4)=1 (and thank goodness that’s the case), then there exist integers x and y satisfying

3x+4y=1

But that isn’t what I wanted!  I wanted something like 3a+4b=5.  But this is easy.  Multiply the above equation on both sides by 5 and we get

5(3x+4y)=3(5x)+4(5y)=5

So just call 5x=a and 5y=b.  So it must be possible.

And people say that pure math has no practical application.