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Six Boxes (game show?)


Question

this a riddle I made up, so i apologize if its too easy or too hard...

There are 6 boxes on the table. According the host: Three are empty ($0). One has $1000, not a bad prize. The fifth has $100,000 and the last has 1 million dollars.

The boxes are A,B,C,D,E and F. You dont know which check (or no check in the case of the empty boxes) is in which box. You do know, however, that E has less than $100,000.

You have the option of selecting 3 boxes and are told how many empty boxes are present in those three boxes. Then you have the option again, for any 3 boxes.

Since it doesn't matter, you choose A, B and C for the first choosing. You are told there are two empty boxes there. For the next pick you choose B, C and D. At the same time, the host takes pity and takes away box F, showing the $1000 check inside, and throws it away. Only ABCD and E remain. You repeat your desire to learn how many empty boxes are among B, C and D. The host says the number, and after a quick thought, you pick a box. The host opens it, showing you a check for $1,000,000.00!

What number did the host say, and what went through your mind?

The number can't be 0, because A can't be two empty boxes by itself. So if the host said 1, it means A is empty for sure, and either B or C is empty, but you have no way of knowing which. There is three empty boxes so either D or E is empty too. There's 4 open choices, with no conclusions. The host couldn't have said 0 or 1.

If the host said 2, then either B and C are empty, or B and D, or C and D. For the last two, it means A is not empty, therefore B and C must be empty because of the conclusion of the first pick. So if the host said two, B and C are empty. Which means A is full, and D is full. And there has to be one more empty box, so that must be E. Therefore if the host said 2, two boxes are known to have a check: A and D.

If the host said 3, it means B, C and D are empty. Duh. This works with the first pick because the two empty boxes of ABC are B and C. Meaning A and E are full... but remember what you know from the beginning? Box E has less than $100,000, and the $1000 box was F and already taken away. Therefore E must be empty.

So the host said there are 2 empty boxes among B, C and D. For the reasons given in that section of the answer spoiler, B,C and E are empty, and A and D have checks.

So how did you know which one, A or D, had a million dollars? That's the catch. Let me quote exactly what I said:

"The host says the number, and after a quick thought, you pick a box."

It never said you knew which one had a million dollars. You either get a million dollars or 100,000 dollars, which is still a ton. You made a random guess between A and D and got lucky.

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