Barge Problem: The profs must be crazy!

This term, you are signed up for Math 10,000—a really, really hard math course taught by both Professor Bloom and Professor Gaugler. Being a diligent student you, of course, (try to) go to class the first week but, oddly, neither professor ever shows up. Hunting around the math department, you discover Bloom and Gaugler in the math lounge. They look awful; neither have shaved, taken a shower, or changed clothes since the start of winter break!

Moreover, they are engaged in the oddest thing. Taped to Bloom’s forehead is the number 9,895,625,675,438 and taped to Gaugler’s forehead is the number 9,895,625,675,439. (They see the other person’s number, but not their own).

To make things stranger, Bloom asks Gaugler, “Do you know your number?” To which Gaugler says, “Nope.” Then Gaugler asks Bloom, “Do you know your number?” To which Bloom says “Nope”. As if stuck in a loop, they continue asking each other this same question and getting the same response over, and over again.

Upon seeing you in the common room, they pause and explain that they have been engaged in this exchange since the beginning of break, with the goal of trying to figure out the number taped to their own foreheads. They tell you that they know the numbers involved are consecutive positive integers. They also explain that they can only ask/answer the question: “Do you know your number?”

Have Bloom and Gaugler gone off the deep end? Will they eventually be able to figure out their numbers using this scheme? If so, roughly when will this happen?

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