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Technical Problems 6

Technical Problems is a weekly column consisting of puzzles and math problems intended to be accessible to undergraduates of all majors. Solutions are posted two weeks later online. 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

A state issues license plates consisting of six digits each between 0 and 9. The state requires that any two license plates differ in at least two places (e.g., the numbers 027592 and 020592 cannot both be used). What is the maximum number of license plates that the state can use?

Problem 2

You are given 40 balloons, the air pressure inside each of which is unknown and may differ from balloon to balloon. You are allowed to choose up to k of the balloons and equalize the pressure in them to the arithmetic mean of their respective original pressures. You can repeat this as many times as you want. What is the smallest k for which it is always possible to exactly equalize the pressure in all of the balloons?

Problem 3

The cells of a 100-by-100 square grid are colored red, blue, green, and yellow in such a way that every row and column contains exactly 25 cells of each color. Prove that there are two rows and two columns such that the four cells which are the intersections of these rows and columns are colored in distinct colors.

Compiled and edited by Matthew Brennan.