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Technical Problems 3

Technical Problems is a weekly column consisting of puzzles and math problems intended to be accessible to undergraduates of all majors. The column features new problems each week as well as solutions to the problems posed two weeks earlier. The solutions to last week’s problems will be included in the column next week. If you are interested in having one or more of your solutions published in the column, please send them to general@tech.mit.edu.

Problem 1

The password to a safe consists of seven different decimal digits. The safe will open if you enter seven different digits and one of them matches the corresponding digit of the password. Can you open this safe in fewer than seven attempts?

Problem 2

A lattice point is a point with all integer coordinates. The two legs of a compass are located at distinct lattice points in the coordinate plane drawn on an infinite sheet of paper. The distance between the two legs cannot be changed. You are allowed to fix one of the legs and swing the other leg to another lattice point. Is it possible to switch the positions of the two legs after a finite number of steps?

Problem 3

Does there exist an infinite sequence a1a2a3, … of 1s, 2s and 3s such that no contiguous block appears twice in a row? For example, neither “33” nor “132132” should appear.

Compiled and edited by Matthew Brennan.