# Math Help - The judge, and 12 boxes

1. ## The judge, and 12 boxes

At the end of a trial an eccentric judge sentenced the accused individual to some number of years in prison. However, the judge pointed out that this individual should 'select' the number of years to serve by himself!
The judge placed 12 boxes on the circumference of a circle. Each box was numbered, the clockwise sequence of numbers on the boxes was: 1, 3, 6, 11, 8, 4, 5, 9, 0, 2, 7 and 10. The sentenced man was told that there is a number of coins in each box; and the clockwise sequence of numbers of coins was 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11. The sentenced man should select a box and the number of coins in the box would determine the length of his sentence.
However, before the man pointed to any box, he asked the judge whether he can get any additional information, namely, how many boxes have numbers which exactly match the number of coins they contain? The judge replied that such information could not be disclosed, as it would allow him to determine the empty box. The man thought a little and pointed to the empty box anyway!
How did he know? Which box did he point to?

2. Is box 7 right?

3. Originally Posted by b0mb3rz
the clockwise sequence of numbers on the boxes was: 1, 3, 6, 11, 8, 4, 5, 9, 0, 2, 7 and 10. The sentenced man was told that there is a number of coins in each box; and the clockwise sequence of numbers of coins was 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11.
That should be clarified, else the way you worded this suggests (box#=coins) :
1=0, 3=1, 6=2, 11=3, 8=4, and so on.

I'm saying this because usually "clockwise" means start at noon position
if nothing is specified to prevent that.

An example could be given to clarify:
Code:
box#  : 1 , 3,  6, 11, 8, 4, 5, 9, 0, 2, 7, 10
coins : 9, 10, 11,  0, 1, 2, 3, 4, 5, 6, 7,  8
Is that correct, or did I miss something?

4. Hi Wilmer

I think clockwise doesn't mean that it must start at noon position but rather that it moves in the direction of the clock does
And your example is exactly what I mean

5. Code:
box#  : 1, 3, 6, 11, 8, 4,  5,  9, 0, 2, 7, 10
coins : 2,*3, 4,  5, 6, 7,  8, *9,10,11,*0, 1
This is the only case where 2 are the same.
There's 2 others where 3 are same, 4 cases with 1 same,
all others have none the same.
So the prisoner picked box#7, as this was the only arrangement that
would give it away if known that 2 was the number.

6. My personal assumption was that the empty box was box 0. Seeing as the judge said that he could not disclose how many boxes had the same box number as number of coins. So logically the rule for all of the boxes is that box number=number of coins within the box.
So if the judge was to disclose the number of boxes which contained the same number of coins as the box number("all of them") the guilty party could easily determine that box 0 was empty.

7. If box 1 contained 1 coin, and that information was disclosed, it would automatically mean the previous box contained 0 coins, which would be box 10 (since it is arranged in a circle). Is that not a correct assertion?

8. It is box 7 (I think).

If you imagine the circle as a clock, where the box's numbers remain constant (That is: Box No. 1 is at 12 O'clock, Box No. 3 is at 1 O'clock, etc...) and the number of coins change as I rotate the circle, the number of coincidences between the box number and the coins it contains will, of course, change. When the judge says that I can deduce the box that contains zero coins by knowing the number of coincidences, he's basically saying that there is a number of rotations that I can make that will produce a unique number of coincidences. If I assume an initial condition of:

Time 12 1 2 3 4 5
Box 1 3 6 11 8 4
Coins 0 1 2 3 4 5

When I apply my first rotation, then the new configuration will be:

Time 12 1 2 3 4 5
Box 1 3 6 11 8 4
Coins 1 2 3 4 5 6

We see that now there's ONE coincidence: Box 1 and Coins 1

By applying 12 rotations (One full revolution), we see the number of coincidences like this:

Rotation - Coincidences
0 0
1 1
2 2
3 0
4 3
5 1
6 0
7 0
8 1
9 1
10 0
11 3

So only by rotating the circle twice do we obtain a unique number of coincidences. Therefore, if we rotate the circle two times from its initial condition, the empty box that contains zero coins aligns itself with Box No. 7.

How did the inmate do all of this in his head after "thinking a little" beats me.

Sorry for my bad English, by the way.