Pencil Grouping Question

Question

There are 20 red pencils, 13 green, 12 yellow and 15 blue. If they are evenly distributed into groups of 3 with different colors, how many groups will you have? This is not a combinations/permutations question, just a straight up how many groups of 3 can you create where the pencils in each group is a different color.

I had to do this by hand but I am wondering if there is a formula for this type of thing. Below is what I did, resulting in 20 groups. I used red all the way through since there were more red pencils than any other color. Then I kept the others colors balanced, rotating between them. Is there a more efficient way to tackle this, like a formula?



start with 20 red pencils, 13 green, 12 yellow and 15 blue

group #1 rgb

leaving 19 red pencils, 12 green, 12 yellow and 14 blue

group #2 rgb

leaving 18 red pencils, 11 green, 12 yellow and 13 blue

group #3 ryb

leaving 17 red pencils, 11 green, 11 yellow and 12 blue

group #4 rgb

leaving 16 red pencils, 10 green, 11 yellow and 11 blue

group #5 ryb

leaving 15 red pencils, 10 green, 10 yellow and 10 blue

group #6 rgy

leaving 14 red pencils, 9 green, 9 yellow and 10 blue

group #7 rgb

leaving 13 red pencils, 8 green, 9 yellow and 9 blue

group #8 ryb

leaving 12 red pencils, 8 green, 8 yellow and 8 blue

......... keep doing this.for a while.....

1 red pencils, 0 green, 1 yellow and 1 blue

group 20 ryb

leaving 0 red pencils, 0 green, 0 yellow and 0 blue

Answer 1

20 13 12 15

-12

__________

8 1 0 3

Best you can do is 13.