## 272. Math Puzzle $1 / 88$

In a mathematics circle, Ilka, Franziska, Susanne, and Katja are practicing solving logical thinking problems. Sebastian challenges them to a game.

He shows them a box with earrings, containing eight individual white ones and two individual red ones. Then he turns to his girls:

"Each of you, I will put two earrings on. You won't see what color they are. Afterwards, you can look at each other.

I will then ask you every minute if you can determine the color of your earrings. The one who knows the correct answer the fastest wins!"

Said and done. After the first minute and even after the second, none of the girls can determine the color of their earrings.

But now, one can see it on her face, Ilka knows the color of her earrings. She calls it out to the others!

How did Ilka come to this decision, and what color were her earrings?
Ilka lists all the possibilities for selecting pairs of earrings (sketch). Since after the first minute none of the girls can determine the color, she knows that the first two variants are ruled out.

(1)

![](https://cdn.mathpix.com/cropped/2024_06_06_69be9c4355ed0dc2557cg-88.jpg?height=362&width=260&top_left_y=1161&top_left_x=772)

![](https://cdn.mathpix.com/cropped/2024_06_06_69be9c4355ed0dc2557cg-88.jpg?height=360&width=122&top_left_y=1165&top_left_x=1064)
![](https://cdn.mathpix.com/cropped/2024_06_06_69be9c4355ed0dc2557cg-88.jpg?height=358&width=258&top_left_y=1164&top_left_x=1208)

Otherwise, at least one girl would have seen the two red earrings and soon known that she herself must have only white ones.

"So," Ilka thinks, "at most one red earring is in play. If I myself have a red one, the other three would see it. Since they are clever, they would then know that Sebastian gave them white ones, as the case of two red pieces has already been ruled out. Since after the second minute none of the girls can still determine the color of their earrings, Ilka calls out: 'So we all have only white ones!'

And she was right!