## Platonic Dice

### Problem 389

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.