Friday Puzzler -- 5 boxes

Time for a hint?

Alex Y

How would you find it if I told you that the $100 started out in one of the even-numbered boxes?

Re: Time for a hint?

Larry Barrett

Start with 3. It will not be there, but will then move to either 3,1, or 5.

Open 3 again. If not there, it will next move to either 2 or 4.

Open 2. If not there, it will next move to either 3 or 5.

Open 3. If not there, it will move to 4.

Open 4.

Re: Time for a hint?

Alex Y

The first two moves seem superfluous, since they get you back to exactly where you started, but your next three moves (or in reverse order) are the correct solution if the money started out in an even-numbered box.

And if not...

Re: Time for a hint?

Larry Barrett

The first two moves are indeed superflous if the $100 starts in an even numbered box, 2 or 4. But the first move did force the next move to be to an odd numbered box, 1, 3, or 5, which is the remaining option. Now open box 3 and the $100 is either there, or will move to an even numbered box. So now follow that sequence: open 2, open 3, open 4 (or the reverse).

So regardless of where it starts, open 3, 2, 3, 4 (or 3, 4, 3, 2) will find the $100.

Re: Time for a hint?

Alex Y

regardless of where it starts, open 3, 2, 3, 4 (or 3, 4, 3, 2) will find the $100.

In the first case, it was initially hidden in 2, then moved to 1, back to 2, and back to 1. In the second case, it went from 4-5-4-5. You never saw it :-(

Hint 2

Alex Y

Once you have made the three guesses that will find the money if it started out in an even numbered box, you will know exactly where the money is.

Re: Hint 2

Larry Barrett

If the $100 starts in an even numbered box, 2 or 4, then opening box 2, then 3, then 4 (or reverse) will find the $100.

If the $100 actually started in an odd numbered box - 1, 3, or 5 - then after your sequence of three moves the $100 will have moved to an even numbered box, then back to an odd numbered box, then back to an even numbered box. So if the first three moves - box 2, 3, 4 (or reverse) - did not find the $100, repeat the same moves.

Correct!

Alex Y

That is correct! Even-odd parity is the key.

So 234234 or 432432 or 234432 or 432234 will all work.

I just realized that you can actually do it in 5 guesses.

After 234, if we haven't found it, we know it is now in an even numbered box, and since we have 4 open, we know it is in 2. Compete the search by opening 3 (which either finds it, or it has moved to 1), then 2. So 23432 and 43234 are solutions.