You have 2 boxes and 20 balls riddle

In summary, to maximize the odds of picking 2 whites on a random selection of 1 ball from each box with 20 balls total (10 black and 10 white), the best distribution would be 5 black and 5 white balls in each box. This is because starting with all the balls in one box or the other would result in a zero chance of success, and any movement of balls into the empty box can only improve the odds. This distribution is similar to other optimization problems involving symmetry.
  • #1
Gokul43201
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Give yourself a minute or two to solve this :

You have 2 boxes and 20 balls, 10 black and 10 white. How would you distribute the balls in the boxes (any number in each box so long as the total is 20) to maximize the odds of picking 2 whites on a random selection of 1 ball from each box ?

i.e: how many blacks and whites in each box ?
 
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  • #2
Well to get the largets chance you want to maximize the numerator and minimize thedenominator of the fraction.

So I guess 5 white and 10 black in one box and 5 white in the other box.
 
  • #3
Actually it'd be better to but 1 white in one box and the rest of the balls in the other I think.
 
  • #4
I think so too...

There's a common trap that a lot of people fall into. Here's the argument I've heard most commonly (including out of my own head). All balls in Box 1 or all balls in Box 2 gives you a zero chance of success. The problem is symmetric w.r.t the boxes, and starting from one of the above extremum cases, and moving balls into the empty box can only improve your odds. So the best odds will the found at the midpoint, i.e : 5 black and 5 white balls in each box.

The "apparent" similarity with problems like maximising entropy in a 2-chamber system, or finding the rectangle of largest area for given perimeter leads to the taking of the bait quite happily
 

1. What is the "You have 2 boxes and 20 balls riddle"?

The "You have 2 boxes and 20 balls riddle" is a mathematical puzzle that challenges the player to find a solution for distributing 20 balls into two boxes, with certain restrictions.

2. What are the rules for solving the "You have 2 boxes and 20 balls riddle"?

The rules for solving the "You have 2 boxes and 20 balls riddle" are that you must evenly distribute 20 balls into two boxes in such a way that no matter how many balls you remove from either box, the remaining balls can be divided equally between the two boxes.

3. What is the best strategy for solving the "You have 2 boxes and 20 balls riddle"?

The best strategy for solving the "You have 2 boxes and 20 balls riddle" is to start with an equal distribution of 10 balls in each box, and then make adjustments as needed to ensure that the remaining balls can be divided equally between the two boxes.

4. Is there only one solution to the "You have 2 boxes and 20 balls riddle"?

No, there are multiple solutions to the "You have 2 boxes and 20 balls riddle". As long as the rules are followed, any distribution of 20 balls into two boxes that allows for equal division of the remaining balls between the two boxes can be considered a valid solution.

5. What is the significance of the "You have 2 boxes and 20 balls riddle" in mathematics?

The "You have 2 boxes and 20 balls riddle" is a classic example of a combinatorial problem, which involves finding the number of possible arrangements or combinations of a given set of objects. It also highlights the concept of even distribution and demonstrates the importance of considering all possibilities when solving mathematical problems.

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