# Optimize a wheel of fortune

(Taken from a mobile game)

You have a wheel of fortune with a total of 30 fields. You start with the following fields (name, size, minimal size, effect):

• "Diamond Field" (D), size 2, min. size 1, get 10 diamonds
• "Coin Field" (C), size 2, min. size 1, get 1.000 coins
• "Unlucky Field" (U), size 26, min. size 5, get nothing

After every turn, you can select a field which will then increase its size by 1 and decrease another field by 1. If you select a diamond or coin field, the unlucky field will decrease.

If one of the fields reached its minimal size, the 3rd one will decrease. If both others are at their minimum, nothing happens. Example:

State  D    C   U
(n)   2D, 23C, 5U  -> Select C -> D will decrease, because U is already at the minimum
(n+1) 1D, 24C, 5U  -> Select C -> D and U are minimal -> nothing happens
(n+2) 1D, 24C, 5U


The costs for turning the wheel are the following:

Turn      1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 ...
Cost [D]  0  1  1  1  2  2  3  3  4  4  5  5  6  6  7  7  8  8  9  9 10 10 10 10 ...


So the turns 21-n will all cost 10D.

You get 10D per day to spend and want to get as many coins as possible. The diamonds that you don't spend can be kept for future days. At the end of a day, the wheel and the cost will reset.

What is the best tactic to convert your diamonds into coins in the long run?

• I assume there's no limit on turns per day? And probability of wheel giving an item is given by "size"/30? – ghosts_in_the_code Apr 29 '20 at 14:59
• Also you start with 0C 0D ? – ghosts_in_the_code Apr 29 '20 at 15:00
• Also can you clarify exactly what and how many choices are that for changing field size. The way I see I see it you have 6 options, 3 choices for which field to increase and for each of these, 2 choices for which to decrease. Which of these 6 choices are legal? – ghosts_in_the_code Apr 29 '20 at 15:06
• @ghosts_in_the_code The limit are the diamonds that you have available, so 10D per day. You always start with 2D, 2C and 26U. If you increase D or C, U will decrease. Increasing U is pretty much useless. – izlin Apr 30 '20 at 6:55
• Increasing U isn't necessarily useless because it's the only way to transfer area from C to D or vice versa. So is it allowed? Also you start with zero coins and zero diamonds on day 1 right? (Apart from the 10 diamonds you get on day 1) – ghosts_in_the_code Apr 30 '20 at 8:33