Little problem(s) 2022.08

[pier4r]

Inspired by this: https://www.youtube.com/watch?v=gNRnrn5DE58 and this.

Someone wants to order a set of 81 rectangular cuboids that can stick with each other, whatever is the face that we pick of the cuboid.

Those cuboids will be used for measurements.

#1
Define the dimensions of those cuboids with the objective to cover the largest number of possible consecutive lengths provided that we give a fixed interval between those.

I am not sure I wrote this properly so I give an example.

Say that I decide to design the 81 elements as cubes with edge of 1 unit (1/1000 of an inch, a millimeter, a digitus or what not); then with 81 of them and an interval of 1 unit, I can cover 81 consecutive lengths: 1,2,3,...,81

I hope the example helps.

#2
Given the devised set, it should be practical for an operator to compose a certain length with maximum 5 blocks. Thus which blocks could be covered combining at most 5 blocks given the 81 designed cuboids? (in general the limit of 5 could be seen as "limit of N blocks").

#3
What are the all possible covered lengths (consecutive or not) that one can assemble given the 81 elements that were designed?

#4
For each possible covered lengths, what is the shortest combination of blocks to equal that length and what is the longest?