# Your \$1,000 Case Incentive | Solution

16.07.2018

Surely, with our power of logic after nine days of working so hard, we should be able to find something in those details that will let us deduce the only possibility, impress our new boss, and earn some good money.

The key thing to work with is that 50% of the time, all jars are mislabeled.

Let’s think about the jar that says, “Apples and Oranges.” When it’s mislabeled, it’s either apples or oranges. If you lift the lid and see both apples and oranges, then you might be done. It’s tempting to think, “then they’re not mislabeled,” but we must think about the logic of that statement: If even one jar is mislabeled, then they would all be mislabeled. Just because the jar that should contain a mixture is labeled correctly doesn’t mean all the jars are labeled correctly. Imagine the shipping warehouse’s embarrassment if they delivered their shipment based on this conclusion, only to find that 50% of the time, they get jars that say, “Apples,” but contain oranges, or vice versa.

So, 50% of the time, you’ll need to check a second jar. If the jar matches, then you know for certain that the other must. If it doesn’t, then you know you need to just swap the “Apples” and “Oranges” lid.

Now, if you check the “Apples and Oranges” lid and you see only apples or only oranges, then you know for certain the jars have been mislabeled.

Here’s where your logic skill comes in.

Suppose the jar contains apples. Now, looking at the jar labeled “Apples,” you know it must contain either oranges or a mixture of the two. But looking at the jar labeled “Oranges,” you know it must also be wrong, so that means it must contain the mixture. This means the jar labeled “Apples” must be oranges. So, you don’t even have to check those two. You can now swap the lids without having to check.

The same goes if you found oranges in the jar. The one labeled “Apples” must be wrong, which means it can’t contain apples, which means it has to contain the mixture, and the one labeled “Oranges” must contain the apples.

You run this by your new boss, and she’s quite impressed. While you haven’t found a way to eliminate the number of checks to one all the time, you have reduced an unnecessary step 50% of the time, so she decides to give you 50% of \$100.

And that’s just the right amount for a night out—or another year of Highbrow membership for you and your friend so you can keep on learning more things to keep you sharp for years to come!.

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