Mar 05, 2022
It's time for this week's puzzler. Now, do you remember the name of the druggist in last week's puzzles George Bailey?
Well, that was just the setup for this puzzle. We thought we'd try another pharmacist puzzle. So it's exactly the same puzzle, with a little twist.
I'll just state this as if there were no puzzle last week. So here it is. You have a drugstore in which you have all these bottles of pills, our shipment of pills has arrived. There are a couple of 100 pills. 300 pills. 400 pills in each but big bottles.
And in last week's puzzle was, you had one bottle of pills where the pills were faulty. They were overweight. Okay, so you had to figure out, with one weighing on your fancy analytical scale, you had to figure out a clever way to weigh the pills and determine at a glance which bottle was the one with the defective and overweight pill.
And you knew that a good pill weighed five grams in a bad pill, an evil pill, weighed six grams. You were told that there was only one bottle that had faulty pills.
And the solution was to take one pill from the first bottle two pills in the second bottle with three pills with that. Okay, so if you were like six grams overweight, you knew that it was the sixth bottle. Two grams overweight, you know it was the second bottle. A very clever solution.
So now you get the telegram that says you might have, let's say you have six bottles of pills. Could be any number but let's say it's six bottles of pills. And the telegram says there could be any number of bottles that have faulty pills.
So the good bottles have pills that weigh five grams each. But either one or two or three or four or maybe all six bottles have pills in them that weigh six grams. And the entire contents of the bottle would have bought the pill, it wouldn't be just one faulty pill in the bottle, it would be an entire right bottle. And remember they would look like the regular pills.
The question is how can George Bailey or anyone do this with one weighing? Or can it be done?
Can it be done with, one weighing, to determine if any of the bottles have defective pills in them? Because it could be anyone bottle that has defective pills or it could be all six of them?, or none, or anything between. Right? So it's a two-part question. A: Can you do this with one weighing? And part 2: How?
Remember last week's puzzler? It was an upgrade to the pharmacy puzzler from the week before. How did the pharmacist's assistant-we're calling him George Bailey-figure out which bottle of pills was defective, using only one weighing on his scale. Something like that!
So here's how he did it. George takes out his analytical scale there, blows off the dust. And he figures out how to do it with one weighing.
Now in the previous puzzler, you took one pill from the first bottle, two pills from the second bottle, three pills from the third, et cetera. Except that won't work in this case, because it's not unique. So here's what he does. He takes one pill from the first bottle, two pills from the second, four from the next, eight from the one after that, then 16, and 32.
So we have six bottles of pills. 1, 2, 4, 8, 16, and 32. So he weighs them. And he's gonna come up with some number that if I was prepared, I would have already added one 1 + 2 + 4 + 8 +16 +32. And I would know, whatever that is, if I multiply that by five, how many grams all of these ought to weigh if they were all correct pills.
But when I find out, for example, that the thing weighs 27 grams more than it should, I can figure out where they came from, because there's only one way to get 27.
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