## the other side of the dice

A simple riddle from the Riddler: if a standard fair six-faced dice is falling on an edge rather than a face, what is the expectation of the sum of the faces sharing this edge?

The solution proposed there is however somewhat convoluted, when the average is simply

$\frac{1}{6}\sum_{i=1}^6 \{i+\frac{1+\cdots+6}{4}-\frac{i+7-i}{4}\}=7$

since the only face that does not share an edge with face i is 7-i…

### 5 Responses to “the other side of the dice”

1. Emmanuel Charpentier Says:

The pictured dice isn’t a standard one to opposite faces do not sum to 7.
You have to ignore the picture to solve the riddle…

2. I thought for a moment that this was a classic 3D perception problem: that cutout diagram doesn’t look like a standard six-sided die. I always thought that opposite sides always sum to the same number (7). But maybe I’m incorrect.

• I did not pay attention (enough) to the picture, which indeed does not depict a standard dice.

3. Do you really need the assumption that opposite faces are $i$ and $7-i$ though?

• No, indeed, you do not, as the sum of the missing/opposite faces is again 1+2+..+6.

This site uses Akismet to reduce spam. Learn how your comment data is processed.