# Mathematics and Statistics

## Problem of the Week: 2014-11-04

### Due Tuesday, November 4 at 5:00 pm

Consider a book with *n* pages, numbered 1 through *n*. The book is divided into three sections, and the sum of the number of digits used in each section is the same. The three smallest values of *n* for which this is possible are *n* = 3, *n* = 6, and *n* = 9. (You cannot do this if *n* = 12; for example the division ending with pages 4, 8, and 12 uses 4, 4, and 7 digits, respectively.)

What is the next value of *n*, after 9, for which you can do this division, and what is the last page of each section?

*n*pages, numbered 1 through

*n*. The book is divided into three sections, and the sum of the number of digits used in each section is the same. The three smallest values of

*n*for which this is possible are

*n*= 3,

*n*= 6, and

*n*= 9. (You cannot do this if

*n*= 12; for example the division ending with pages 4, 8, and 12 uses 4, 4, and 7 digits, respectively.)

*n*, after 9, for which you can do this division, and what is the last page of each section?

All answers should be clearly explained. Mathematics is about logic and reasoning, not just about getting a number and putting a box around it.

Submit your answers to the Math/Stat Office, Room 107, Adel Mathematics Building. If your instructor is giving you extra credit for doing POTW, please indicate the instructor's name and the class. Please put the due date (e.g. 2014-01-21 for January 21, 2014) on your submission. Contact Jim.Swift@nau.edu with questions.

Winning solutions and a summary of scores will be posted on the POTW bulletin board. This bulletin board is located near the bottom of the big stairway joining the first and second floor of the Adel Math Building.