You are the cargo director on the maiden voyage of the S.S. Buoyant, and you’ve agreed to transport several tanks containing the last specimens of a critically endangered fish species to their new aquarium. Unfortunately, as you’re passing through shark-infested waters, the boat is battered by a fierce storm, throwing your precious cargo overboard. And to make matters worse, no one seems certain just how many fish tanks are missing. Fortunately, you have a rescue sub at your disposal, but only enough fuel for one trip to the ocean floor. You need to know where the tanks are so you can gather them all in one quick pass.
Not a single fish can be lost. You decide to scan the three sectors of the ocean floor where the cargo could have landed. Thermal imaging shows 50 organisms in the area, and you quickly realize that that number includes both your fish and some ravenous sharks. You flip on the sonar to get a better look. The image for Sector Alpha shows four tanks and two sharks, the image for Sector Beta shows two tanks and four sharks, and the image for Sector Gamma is blank.
Your sonar has malfunctioned, and you’re going to have to go with the info you have. You check the shipping notes, but all you learn is that each tank had the same number of fish inside. The cargo hold had space for anywhere from 1 to 13 total tanks. And finally, the old captain tells you that this area has the odd property that no two sectors can have the same number of sharks, but every sector will have at least one, and no more than seven. There’s no time to waste. The tanks won’t withstand the pressure much longer.
As you descend in the sub, you review everything you know. How many fish tanks do you need to find in Sector Gamma? Hurry, the fate of an entire species depends on you. Pause here if you want to figure it out for yourself. Answer in: 3 Answer in: 2 Answer in: 1 At first, it seems like there are just too many missing pieces of information. After all, you don’t know how many fish or how many tanks there are, let alone how many fish are in each one. But then you remember the best way to compare multiple pieces of partial information – a table. Since we know there are thirteen tanks at most, and we already see six tanks in Sectors Alpha and Beta, we know the total number of tanks must be between 6 and 13.
We also know that each sector has a different amount of sharks with no more than seven in each one. Since there are two in Sector Alpha and four in Sector Beta, Sector Gamma can have 1, 3, 5, 6, or 7 sharks. What about the number of endangered fish? Out of the 50 total organisms in all three sectors, we know at least seven are sharks, leaving a maximum of 43 fish inside all the tanks. And the more sharks we find in Sector 3, the fewer fish there are to save.
Now, remember that the fish are equally distributed across all the tanks. Why is that important? Because it means that one of the possible values for the total amount of fish must be divisible by one of the possible values for the total amount of tanks. And looking at the table, we can see that the only combination that works is 39 fish divided between 13 tanks with three fish in each.
With sharks swarming around, you quickly pilot the sub through the first two sectors before retrieving the remaining seven tanks in Sector Gamma. You’ve saved the species and taken an impromptu dive. All in all, not a bad day, unless you happen to be a hungry shark.