## Loaded Dice

Posted by Karl Sharman on September 21, 2010 – 9:18 am

You have a single six sided die with one integer on each side. The integers are random positive numbers and do not repeat.

When the die is rolled, it is placed against a wall, which means 4 sides are visible, the front, the top and the two sides, and the bottom and back are hidden from view.

The die is rolled 3 times, and the person rolling the die calls out 2 values for each roll, the sum of the front and top numbers, and then the sum of all 4 visible sides.

After the following 3 rolls, what numbers are on the die in what possible arrangement?

Roll #1 = 18, 28

Roll #2 = 7, 18

Roll #3 = 6, 22

September 21st, 2010 at 10:06 am

One possible solution is that the die has 2, 4, 5, 7, 8, & 11 on its sides.

Opposite sides contain the following pairs:

2 & 8

4 & 7

5 & 11

The die would look like the following if you were to flatten it.

7

11 2 5

4

8

First roll (18, 28) has the 11 on top, 7 in front, 2 on left and 8 on right.

Second roll (7, 18) has the 2 on top, 5 in front, 4 on left and 7 on right.

Third roll (6,22) has the 2 on top, 4 in front, 11 on left and 5 on right.

September 21st, 2010 at 10:08 am

Sorry, my previous post did not have the die layout appear correct. The 7, 4, & 8 should appear above and below the 2. I tried to use spaces to make the layout look like a t or a cross.

Here is a better representation of the die.

7

2 5 11

4

8

September 21st, 2010 at 3:19 pm

John24 is right. I will just add that this is THE only possible solution of the problem

September 22nd, 2010 at 12:42 am

Remarks: It is not necessary to say in the problem that the sides of the dice are different numbers – there is no extra solution if some of them are equal.

If we don’t consider integer sides, but release the restriction to just positive real numbers, then there is another (but that’s all!) solution: (3.5, 6.5) (0.5, 15.5) (2.5, 8.5), where I have paired the opposite sides of the dice, which is enough to fully characterize it.

It might be an interesting problem to really involve probability here. What I mean is – for the outcomes of the three rolls we use a pair of randomly generated integers (the smaller one for the sum of the two neighboring sides, and the bigger one for the sum of all the four sides) between 1 and say 30. What is the probability that such a dice exists? (I still haven’t thought about it, so I don’t know if there is an easy, elegant solution to the problem!)

September 23rd, 2010 at 10:09 am

John 24 scores this one very quickly!

You can determine the die’s opposite sides from the rolls, but not the arrangement of them. Specifically, one set of opposite sides can be the mirror image of the other.

The opposite sides are

2&8, 7&4, 5&11