I’m a graduate student in biology that encountered a problem that I really struggled solving in my project. I solved it using an iterative approach (will describe more below) but was curious if there was a simpler or more elegant solutions to this.

I wanted to generate a sequence of numbers. Let’s assume we have vector v = 1:12. I want to generate a sequence consisting of the vector v = 1:12, with n repetitions. The resulting sequence will be of length v * n. Additionally, for each element of this sequence, I want to add a boolean value (I will explain more below).

I want this sequence to meet several conditions:

1. In each repetitions of vector v, I want the elements 1:12 to appear in random order. (this seems straightforward enough to me)
2. The boolean that is being assigned to each element of the sequence must be balanced within a repetition. That is, for a given nth repetition, I want there to be 6 true and 6 false.
3. Each value in vector v should have the same number of true and false across the entire sequence.

For example if n=10, I have a sequence of length 120. In this sequence, there should be 1 with 5 true and 5 false. 2 with 5 true and 5 false so on and so forth.

I tried to solve this problem by imposing conditions. If we get really unlucky when generating the initial repetitions, by n/2 repetition, we may have a value in v that never gets a true draw. Then, I need to place a condition that this value needs to be drawn for the remainder of the repetitions etc etc. You can apply similar logic for a value that has only been drawn once, twice, etc..

What would be the most easiest way to solve this? I have a code in MATLAB that works but would love to hear more solutions to this. It reminds me of a sudoku problem but honestly I have to admit I spent way too much time trying to figure this out so am reaching out for help. Thanks!