Question

How many glasses will remain empty if you half-fill some of the empty glasses, from the given set of filled glasses? 

 

Hint:

Word puzzles often hinge on precise definitions. The key here is realizing that a glass you pour from can become empty, and a glass you pour into is no longer empty. When an answer is known, working backward can be a great way to uncover the intended logic of the puzzle.

Answer 1

Step 1: Understanding the Concept: 
This is a logical puzzle that plays on the definition of "empty" and requires careful tracking of the state of the glasses. We first need to determine the initial number of filled and empty glasses. 

Step 2: Detailed Explanation: 
Initial Count: 
- By observing the image, we can count the total number of glasses. There are 4 rows and 6 columns, so \(4 \times 6 = 24\) total glasses. 
- The shaded (filled) glasses can be counted: 2 in the front row, 3 in the second, 3 in the third, and 2 in the back row. Total filled glasses = \(2+3+3+2 = 10\). 
- The number of empty glasses is therefore \(24 - 10 = 14\). 
- So, we start with 10 filled glasses and 14 empty glasses. The Action and its Consequences: 
The question asks what happens if we "half-fill some of the empty glasses, from the given set of filled glasses". 
- When we take water from a "filled" glass, it is no longer full. To get the water, we must empty some of the filled glasses. 
- When we pour water into an "empty" glass, it is no longer empty; it becomes "half-filled". 
Logical Deduction: 
Let's model the process to arrive at the answer of 10 remaining empty glasses. 
Let \(x\) be the number of filled glasses that we decide to empty completely to perform the task. 
The water from these \(x\) filled glasses can be used to half-fill \(2x\) empty glasses. The glasses that are empty at the end are: 
1. The filled glasses that we emptied: \(x\). 
2. The originally empty glasses that were never touched: \(14 - 2x\). 
Total number of empty glasses at the end = \(x + (14 - 2x) = 14 - x\). The question implies a specific scenario has occurred. The answer is given as 10. Let's see what value of \(x\) gives this result. 
\[ \text{Total Empty Glasses} = 10 \] \[ 14 - x = 10 \] \[ x = 4 \] This means the scenario is that we took 4 of the filled glasses and used their contents. Let's verify this scenario: 
- We start with 10 filled and 14 empty glasses. 
- We take 4 filled glasses and empty them. These 4 glasses are now empty. 
- The water from these 4 glasses is enough to half-fill \(4 \times 2 = 8\) of the empty glasses. 
- So, 8 of the original 14 empty glasses are now half-filled. 
- The number of untouched empty glasses is \(14 - 8 = 6\). 
- The total number of empty glasses in the end is the sum of the glasses we emptied and the glasses that were untouched: \(4 + 6 = 10\). 
This confirms the logic. 

Step 3: Final Answer: 
If we empty 4 of the filled glasses to half-fill 8 of the empty glasses, we are left with 10 empty glasses.