Lots of ground covered today. So as to maximize the opportunity for students (and parents) to sink their teeth into the problem we ended with, I'll forego all the things we looked at earlier in the lesson and cut to the chase. I revived a problem some of the students had looked at with Jason. A version of this appeared in the third Die Hard movie (the one with Samuel L. Jackson as a locksmith), in which Bruce Willis has to figure out the answer in order to get another clue towards preventing a public school building from being blown up (it's an elaborate bluff/distraction from plans for a huge robbery): you must fill a jug with exactly 4 gallons of water given only a 3 gallon and a 5 gallon jug and an unlimited supply of water. You don't get any measuring tools or markers: just the two jugs and the water.
This is, by now, a very well-known problem, and yet many people find it extremely challenging to figure out. It turns out that it doesn't matter whether you start by filling the 3-gallon jug or the 5-gallon jug: either approach can lead to a solution.
I reposed that problem today only with a 5-gallon and 7-gallon jug. Further, rather than asking students to make one particular number of gallons, I asked them if it is possible to measure exactly every whole number amount of water from 0 to 12 gallons, inclusive. That is where class ended, with most students well on their way to a solution. I then asked them to consider for tomorrow whether this is possible with any two jugs that hold different whole numbers of gallons; that is, for jugs holding A and B number of gallons (both whole numbers), is it always possible to measure precisely all whole numbers from 0 to (A + B) inclusive. If not, what must be true about A & B for this to be possible?
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