Bernardo and Sylvia Play a Game

Task

Bernardo and Silvia play the following game. An integer between 0 and 999, inclusive, is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds 50 to it and passes the result to Bernardo. The winner is the last person who produces a number less than 1000. What is the smallest initial number that results in a win for Bernardo?

IM Commentary

This task presents a simple but mathematically interesting game whose solution is a challenging exercise in creating and reasoning with algebraic inequalities. The core of the task involves converting a verbal statement into a mathematical inequality in a context in which the inequality is not obviously presented, and then repeatedly using the inequality to deduce information about the structure of the game.

In addition to content standards concerning the creation and use of inequalities, the task serves as an illustration of mathematical practices: For one, the task requires the ability to persevere through repeated applications of the inequalities, and to make sense of the results in the context of the game (MP1). The recursive process of working backwards from a desired result involves considerable abstract and quantitive reasoning (MP2) applied to the structure of the game (MP7). Recognition that the game situation can be represented using inequalities requires some facility with mathematical modeling (MP4).

This task was adapted from problem #20 on the 2012 American Mathematics Competition (AMC) 10B Test.

For the 2012 AMC 10B, the question was "Let $N$ be the smallest initial number that results in a win for Bernardo. What is the sum of the digits of $N$?"

For the 2012 AMC 10B, which was taken by 35,086 students, the multiple choice answers for the problem had the following distribution:

Choice	Answer	Percentage of Answers
(A)*	7	21.93
(B)	8	8.74
(C)	9	10.69
(D)	10	7.99
(E)	11	6.99
Omit	--	43.63

Of the 35,086 students: 17,169, or 49%, were in 10th grade; 9,928 or 28%, were in 9th grade; and the remainder were below than 9th grade.

Solution

The smallest initial number for which Bernardo wins after one round is the smallest integer solution of $2n+50\ge1000$, which is $475$. The smallest initial number for which he wins after two rounds is the smallest integer solution of $2n+50\ge475$, which is 213. Similarly, the smallest initial numbers for which he wins after three and four rounds are 82 and 16, respectively. There is no initial number for which Bernardo wins after more than four rounds. Thus the smallest integer where Bernardo wins is 16.