You have two hourglasses: one measures 7 minutes and the other measures 4 minutes. You need to measure exactly 9 minutes using these two hourglasses. You can flip the hourglasses as many times as needed, but you must use them to measure exactly 9 minutes.

Objective:
Determine the sequence of flips and timing to measure exactly 9 minutes using the 7-minute and 4-minute hourglasses.

Solution

The key to solving this puzzle is to use both hourglasses simultaneously and strategically flip them to track the passage of time.

Step-by-Step Solution for Hourglass Puzzle:

Step 1: Start Both Hourglasses Simultaneously

• Flip both the 7-minute and 4-minute hourglasses at the same time.

Step 2: Track the First 4 Minutes

• When the 4-minute hourglass runs out, 4 minutes have passed. At this point, the 7-minute hourglass will have 3 minutes remaining.

Step 3: Flip the 4-Minute Hourglass

• Immediately flip the 4-minute hourglass to start it again.

Step 4: Track the Next 3 Minutes

• When the 7-minute hourglass runs out (after another 3 minutes), a total of 7 minutes will have passed. At this point, the 4-minute hourglass will have 1 minute of sand remaining.

Step 5: Flip the 7-Minute Hourglass

• Immediately flip the 7-minute hourglass to start it again.

Step 6: Track the Last 2 Minutes

• After 1 minute (when the 4-minute hourglass runs out again), exactly 8 minutes will have passed in total. The 7-minute hourglass will now have 6 minutes of sand remaining.
• Now, flip the 4-minute hourglass again.

Step 7: Measure the Final Minute

• After 1 more minute (when the 4-minute hourglass runs out for the last time), exactly 9 minutes will have passed.

Final Answer

By carefully flipping the hourglasses according to the steps outlined above, you can measure exactly 9 minutes.