1. ## Problem Solving

Three bags:Each bag contains 2 apples. 2 green in one, 2 red in another and one of each colour in third bag. All 3 bags are mislabelled. You may look at only one apple from any one of the bags. Which bag should you select from to be sure you can determine the content of all three bags.

2. $\displaystyle \sqrt{2.25}$

You know that $\displaystyle 2.25 = \frac{225}{100}$

$\displaystyle \sqrt{\frac{225}{100}}$

$\displaystyle \frac{\sqrt{225}}{\sqrt{100}}$

I think you can do it now.

3. Hello, not happy jan!

This is a classic (very old) problem . . .

Three bags: each contains two apples.
Two green apples in one, two red apples in another, and one of each colour in the third.
All 3 bags are mislabeled. You may look at one apple from any one of the bags.
Which bag should you select from to determine the contents of all three bags.

The key is that the bags have the wrong labels.

The bags look like this: .$\displaystyle \boxed{\text{G/G}} \quad \boxed{\text{R/R}} \quad \boxed{\text{G/R}}$

Take an apple from the bag labeled $\displaystyle \boxed{\text{G/R}}$

Suppose it is Green. .
We can reverse the argument later.
Since it is not the G/R bag, it must contain $\displaystyle \text{G/G}$.

. . We have: . $\displaystyle \begin{array}{c|c|c|c|} \text{Labels} & \boxed{\text{G/G}} & \boxed{\text{R/R}} & \boxed{\text{G/R}} \\ \hline \text{Contents} & & & \text{G/G} \end{array}$

Where is the $\displaystyle \text{R/R}$ bag?
Since it is not the one labeled $\displaystyle \boxed{\text{R/R}}$, it is the one labeled $\displaystyle \boxed{\text{G/G}}$

. . We have: . $\displaystyle \begin{array}{c|c|c|c|} \text{Labels} & \boxed{\text{G/G}} & \boxed{\text{R/R}} & \boxed{\text{G/R}} \\ \hline \text{Contents} & \text{R/R} & & \text{G/G} \end{array}$

Finally, $\displaystyle \boxed{\text{R/R}}$ contains $\displaystyle \text{G/R}$

. . Therefore: . $\displaystyle \begin{array}{c|c|c|c|} \text{Labels} & \boxed{\text{G/G}} & \boxed{\text{R/R}} & \boxed{\text{G/R}} \\ \hline \text{Contents} & \text{R/R} & \text{G/R} & \text{G/G} \end{array}$

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

This problem can be made even more baffling like this . . .

. . One bag contains 11 green apples.
. . One bag contains 13 red apples.
. . One bag contains 7 green and 8 red apples.

The bags are labeled: .$\displaystyle \boxed{\text{Green}},\;\boxed{\text{Red}},\;\boxed {\text{Mixed}}$
. . but each bag is incorrectly labeled.

You may draw one apple from any bag,
. . examine its color, and set it aside.
You may repeat this sampling process.

What is the least number of samples you must take
. . to determine the contents of each bag?

The solution is the same:
. . Take one from the bag labeled "Mixed".