## Platonic Dice

### Problem 389

Published on Saturday, 16th June 2012, 05:00 pm; Solved by 831; Difficulty rating: 30%
An unbiased single 4-sided die is thrown and its value, `T`, is noted.

`T` unbiased 6-sided dice are thrown and their scores are added together. The sum, `C`, is noted.

`C` unbiased 8-sided dice are thrown and their scores are added together. The sum, `O`, is noted.

`O` unbiased 12-sided dice are thrown and their scores are added together. The sum, `D`, is noted.

`D` unbiased 20-sided dice are thrown and their scores are added together. The sum, `I`, is noted.

Find the variance of `I`, and give your answer rounded to 4 decimal places.