In my most recent, non-reported, second problem (the first one was heavily reported two-four years ago which discouraged me from making problems), there was a change made to it by the staff which got me thinking (they added that the die were unbiased), what if the die were biased but equally, can Alex get a higher expected score (Xela uses a copy of the die Alex uses so the probabilities of both die are equal), turns out it’s true, he can get a higher expected score but since I do not know how to do multi-variable calculus, I couldn’t find the upper bound
Defining some terminology: = Probability of i appearing on the dice,
Chance of getting out
Average score per round
Here’s a picture to help you see how I got these formulae
Using this program I was able to find somewhat where the answer should lie
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
After more than 1 billion random checks, the highest expected score I reached was 23.752142249471124 with 0.1487492948991272, 0.14899405525187628, 0.154656034331149, 0.16515216949050074, 0.1808399982323611, 0.20160844779498574, which is not very far from the 23.33… obtainable from an unbiased die
If there are any mistakes in the formulae I made please tell.
If there are any improvements I can make to the program to make it faster or if u got a higher expected score than me, please put it down with the probability distribution
Note: If you want to know what’s up with the number of cosines and sines in the program, it’s to get a fair distribution of all possible probability distributions of the die, the probability distribution represents points on a six dimensional cube with its diagonals as the axes(points are taken from only one face as I require only positive numbers), and the list “a” contains the angles to reach that point in a circular co-ordinate system.