Four, (20 points) In a box, there are a total of 88 small balls in three colors: red, yellow, and black. It is known that by randomly taking out 24 balls, it can be guaranteed that at least 10 of the balls are of the same color. Under the condition that this is true, regardless of how the balls of each color are distributed, what is the minimum number of balls that must be randomly taken out to ensure that at least 20 balls are of the same color?
First, prove that taking out only 43 balls is not enough.
In fact, when there are 42 red balls, 41 yellow balls, and 5 black balls in the box, if you take any 24 balls, then the number of red and yellow balls is at least $24-5=19$, so there must be at least 10 red or yellow balls; but taking 19 red balls, 19 yellow balls, and 5 black balls, a total of 43 balls, there are no 20 balls of the same color.

Next, prove that taking 44 balls from the box, there must be 20 balls of the same color. Let the number of red, yellow, and black balls in the box be $x$, $y$, and $z$ respectively, and assume $x \geqslant y \geqslant z$.
(1) If $z \leqslant 5$, then among the 44 balls taken out, the number of red and yellow balls is at least $44-5=39$, so there must be at least 20 red or yellow balls.
(2) If $z=6$, when $y \leqslant 8$, the number of red balls among the 44 balls is greater than or equal to $44-8-6>20$; when $y \geqslant 9$, taking 9 red balls, 9 yellow balls, and 6 black balls, there are no 10 balls of the same color among the 24 balls, which does not meet the conditions of the problem.
(3) If $z=7$, when $y \geqslant 8$, taking 9 red balls, 8 yellow balls, and 7 black balls, there are no 10 balls of the same color among the 24 balls, which does not meet the conditions of the problem; when $y=7$, the number of red balls among the 44 balls is greater than or equal to $44-7-7>20$.
(4) If $z \geqslant 8$, then taking 8 red balls, 8 yellow balls, and 8 black balls, it is known that the conditions of the problem are not met.

In summary, at least 44 balls must be taken out to ensure that there are 20 balls of the same color.
(Xiong Bin, Li Deyuan, Gu Hongda, Liu Hongkun, Ye Shengyang)