This is a genuine question and I don't know the "correct" answer to this. I'm also of two minds if it is really on-topic on the site. Finding the answer is surely "a puzzle" (if possible), but it might actually also be impossible? In worst case, this post can serve as a generalized post for a group of puzzles like f.e. this, this, or this. Please be gentle and just turn a blind eye if you find this puzzle offending. (Ther story is fictional.)
As the chef-designer of a British dice-making company, you have been given the task to create the ideal set of dice for Scrabble. Your boss is not really into nitty-gritty details, he just wants something nice as a product. After some thought, you've decided that the following restrictions and conditions apply:
The set can only consist of dice shapes, which are produced by the company. These are the five platonic solids (D4, D6, D8, D12, D20) and a ten-sided die (D10).
A "dice roll" should produce an outcome as closely matching the letter frequency table for English, e.g. the likelihood of getting a particular letter should best possible match this distribution.
(In other words: Doing a lot of "dice rolls" should produce this distribution.)
The set may consist of as many dice as needed, but the lower the number of dice, the better the set can be sold.
(Packaging costs for an infinite set of dice are, well, rather high...)
Each side of any die may only show letters (A-Z) or a blank. No "wildcards", symbols of special meaning etc. However, a side may show more than a single letter.
No complicated "evaluation" rules allowed. At any "dice roll", the outcome set of letters is the sum of all letters shown on the dice faces.
A single "dice roll" is defined as rolling all N dice of the set and producing M letters
As each "dice roll" produces M letters (follwing the distribution of the letter frequency table), the dice set with the lowest M is preferred.
(It is hard to build a word with too many letters, but easy to allow multiple rolls for a word...)
What dice set would you propose?
The "winner" here is the most practical set. It should reproduce the frequency table in good approximation and allow "rolling up a few letters" with relative ease.
Bonus, if a method of constructing the set is provided, allowing it to be customized to different languages, i.e. different frequency tables.
However, the answer must contain some simple to follow instructions on how to produce the set, suitable to be passed on to your boss and colleages at production. (Who are all down-to-earth men with few mathematical skills.) So, while mathematical deductions and proofs are welcome, there also needs to be a "final" example set of dice listed explicitly. (i.e. Number of dice, their type and what is on their faces. )
The puzzle was inspired by this die:
It is a nice die, but rolling it N times will give you N letters of even distribution (and the wildcards), so I was wondering for something better than that, which could be really used in Scrabble. And then I realized, that different languages would need different sets due to their different letter frequencies... And the puzzle/question was born.
Results so far:
Compiled from the answers below, the following distributions have been achieved:
Link to results table (image)