A Missing Key

Since only 1 of the statements is false , the other 2 should probably be true :

If the 1st statement is false , then statements 2 and 3 should be true , which is alright since there is no contradiction.

If the 2nd statement is false , then statements 1 and 3 should be true . But the 2 statements(1&3, that are supposed to be true) are contradicting each other, making them wrong.

If the 3rd statement is false , then statements 1 and 2 should be true .But the 2 statements(1&2, that are supposed to be true) are contradicting each other, making them also wrong.

Since we know that the 1st Statement is the False Statement , the remaining two statements are True :

The key is in Box 1: False

The key is in Box 2: True

The key is not in Box 1: True

Thus, the key is in the 2nd Box

The first and third cannot both be true as they contradict each other. The first and second cannot both be true as they also contradict each other. Therefore, the third and second are true. The second statement says Box 2 has the key, therefore the key is in box 2.

Looking at statement 1 and statement 3, we can understand that they are opposing each others statements. So, we cannot comment above the honesty of these two statements efficiently(because 1 should be false). The second statement shows a more likely statement. There is no other statement opposing it. Now, what about box 3? Nothing is mentioned about box 3. But, only one of the statements given is false, so in this case, two should be true. We have proved that either statement 1 or statement 3 is false, that leaves only 2. So, we should not consider box 3.