# When do we begin?

**Logic**Level 4

I have two hourglasses. One measured 9 minutes, and the other measured 22 minutes. If I want to measure 33 minutes, what is the minimum number of times I need to turn the hourglasses?

**Details and Assumptions**:

- As an explicit example, suppose I want to measure 4 minutes. Then I will have to turn both hourglases at the start. At the 9th minute, I will flip the 9-minute hourglass again. And when the same hourglass is finished, which is at the 18 minute mark, I would start the interval of the 4 minute interval. When the 22-minute hourglass is finished. I will have measured \(22-18=4\) minutes interval. Which gives a total of 3 turns of the hourglass, 2 from the 9-minute hourglass, and 1 from the 22-minute hourglass.

**Bonus**: Prove that we can measure all the integral minute interval using these two hourglasses alone.