6-dimensionarians have a 9-letter alphabet (consisting of all letters from a to i inclusive). Their language consists of 3-letter words, such that there is no 2 identical letters in any 2 of them. Their words for a big bad dog look strangely familiar: big bed dag. How many are the remaining words of the 6-dimensionarians?
Today I received an e-mail from the Ministry of External of Heirs:
We highly recommend that, instead of posting them publicly, you should send national-security-related questions directly to the Ministry. The language of the 6-dimensionarians consists of the following 10 words: big, bed, dag, abc, aef, agh, bfg, cdf, ceg, chi. Although we have no idea how you obtained such a classified information about the only partner-race of ours living in the 6-th dimension, we have to admit your knowledge makes you one of the top-experts on interdimensional communication. Therefore, we have a question for
How many languages are there in the Universe that share the same:
> letters (from a to i inclusive),
> word-formation rules (number of letters in a word equals 3, number of identical letters in any 2 words equals 1) and
> first 3 words (bid bed dag),
with the language of the 6-dimensionarians, and whose other words have the same
letters (although some/all of them are in a different order)?
Robot-assistant to the Minister
My answer to the Ministry was short:
Imagine a 7-dimensional cube, the edge of which is 3! units long (let’s call it big-bed-dag cube). Imagine now a 3-dimensional cube, the edge of which is 1 unit long (let’s call it small cube). To answer your question is to answer the question how many small cubes the big-bed-dag cube consists of. The big-bed-dag cube represents all the languages in accordance with your definition, one of which is the language of the 6-dimensionarians (any other similar language being represented by a small cube too). The total number of languages is (3!)^7, which is exactly 279,936. The number of those that are different from the language of the 6-dimensionarians is therefore 279,935.