On the board game of Amun Re: The Card Game there is this rule that asks you to combine a certain number of coin cards until you have a specific amount of money.
For example with Card Coins of (0, 1, 2, 3, 4, 5, 6, 7, 8) you are asked to select a number of cards that total up to a specific amount (something smaller than 40).
Very quickly I got tired of running the combinations on my head on what specific cards I should keep if I had to have a total of, for example, 18 coins (there are 14 possible sets) so I wrote a small script to generate all possible combinations until a total sum of 40 (actually you cannot reach sums larger than 36, since you do not have that many card coins)
This resulted to this table.
By looking at the combination of cards, I wondered "which card appears more often in these, seemingly, random sets"?
After changes to my script to count the number of occurrences I found something surprising:
{0: 255, 1: 128, 2: 128, 3: 128, 4: 128, 5: 128, 6: 128, 7: 128, 8: 128}
I would have never thought that the number of occurrences is the same for every card (well, except 0 that appears in all sets).
Running the same, for different initial card coins, I found the exact same pattern:
For sequences of cards running up to 9:
{0: 511, 1: 256, 2: 256, 3: 256, 4: 256, 5: 256, 6: 256, 7: 256, 8: 256, 9: 256}
For up to 10:
{0: 1023, 1: 512, 2: 512, 3: 512, 4: 512, 5: 512, 6: 512, 7: 512, 8: 512, 9: 512, 10: 512}
etc
How can that be? What is the underlying principle that "forces" each card to appear the same number of times?