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Technical Problems 4

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. Solutions to the problems posed on April 9 are posted on our website. 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

Initially there are 111 pieces of clay on a table, all with equal mass. In a single move, you are allowed to take several groups of pieces of clay, such that each group contains an equal number of pieces of clay, and combine the pieces in each group into a single piece of clay. What is the smallest number of moves required to end up with 11 pieces of clay, all of which have different masses?

Problem 2

A multi-digit positive integer is written on a blackboard. In a single step, you are allowed to put a number of plus signs between some pairs of adjacent digits. The addition is performed and the process is repeated with the sum. For example, given 123456, if you write 123 + 4 + 56, then the resulting number is 183. Show that regardless of what number was initially on the blackboard, you can always obtain a single-digit number in at most ten steps.

Problem 3

There are some markets in a city. One-way streets join some pairs of these markets such that exactly two streets leave each market. Prove that the markets in the city can be partitioned into 1014 districts such that no two markets from the same district are joined by a street and such that the streets between any two districts all point in the same direction.

Compiled and edited by Matthew Brennan.