Consider two problems you’ve probably heard before:<\/p>\n
These problems are clearly related, but how? Well, I’ll tell you how — by the good old GCD. Here’s my method for solving these problems:
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The last step is the hardest to understand. Let’s start with the buckets. We interpret a positive bucket as filling a bucket with water, and a negative bucket as pouring one out. So if we use the linear combination 3 * 3 – 1 * 5 = 4, we will have to fill 3 three-gallon buckets with water, and pour out 1 five-gallon bucket. Specifically:<\/p>\n
That’s fairly easy to understand, because it’s easy to interpret a “negative bucket” as pouring out water. For the hourglass problem we have to figure out how to measure “negative time.”<\/p>\n
We’ve got a five-minute hourglass and a three-minute hourglass. We can subtract them to make two minutes: turn them both over at the same time. When the three-minute one is done, there will be two left in the five-minute one. If we then flip over the three-minute timer and wait until the five-minute one finishes, flip over the three-minute one to get two additional minutes. See what we did there? We measured the difference between the three-minute timer and the five-minute timer inside the three-minute timer.<\/p>\n
How does this apply to our problem? We’ve got 2 * 5 – 1 * 3 = 7. Let’s separate that as 1 * 5 + ( 1 * 5 – 1 * 3) and interpret this as “Start the five-minute timer and three-minute timer at the same time. Take the difference in the three-minute timer and add it on at the end.” Easy-peasy!<\/p>\n
Challenge questions (all apply to both hourglasses and buckets):<\/p>\n
Consider two problems you’ve probably heard before: Given a five-gallon bucket and a three-gallon bucket, measure exactly four gallons of water. Given a five-minute hourglass and a three-minute hourglass, measure exactly seven minutes. These problems are clearly related, but how? … Continue reading