Given a list from 1 - 12, assuming I use each number every 10 minutes, how do I maximize the number of minutes between close numbers.
In other words, maximize the difference between each value in each spot of the list.
Trying to maximize the time between n & n+1, but also n & n+2, and n & n+3
1 next to 2 is the worst. 1 next to 12 would be best, however it would end up causing closer numbers further down the list.
For example,
Is there an optimal Delta between each that could be calculated given the number of items in a list?
Assuming that the input values in the set are evenly spaced, I would recommend using the following approach to define the sequence.
Step through the ordered values, jumping X number of values each time, where X is defined as the total number of values / Phi. Pretend that the end of the set of values circles back to the beginning.
So for the set of values 1 - 12, you'd have:
X = 12 / Phi
X = 12 / 1.618 = 7.4
Round off 7.4 to the nearest integer, so let's assume X = 7.
Then your sequence would be 1, 8, 3, 10, 5, 12, 7, 2, 9, 4, 11, 6
To quantify (or score) how "maximized" this sequence is, you would take the sum of the following calculation for EVERY member in the set.
For EACH set member, calculate the absolute value of the difference in "value" between itself and each other member divided by that "distance" to that member. For example, for the member 8 in the above sequence, its score would be:
8,1 = |8-1| / 1 = 8
+
8,3 = |8-3| / 1 = 5
+
8,10 = |8-10| / 2 = 1
+
8,5 = |8-5| / 3 = 1
+
8,12 = |8-12| / 4 = 1
+
...
Do that for each member of the set and take the sum to get the overall "score". The higher the score, the more "maximized" the sequence will be.