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Mysterious Numbers


Problem Statement
The number 142, 857 is an odd number, but it is also a mysterious number. Multiply this number by 2 through 7 and see why it carries mystery. What is the reason for this? Are there other numbers that may be considered “mysterious?”

Problem setup

I wrote the number 142,857 six times down the page along with x 2, x 3, etc.  


Plans to Solve/Investigate the Problem

I planned to multiply the numbers on the calculator and write them down. I assumed I would see an interesting pattern.


Investigation/Exploration of the Problem

This is either a very simple problem or I am oversimplifying it. The pattern is mysterious: 142, 857 x 2 = 285, 714

142, 857 x 3 = 428, 571

142, 857 x 4 = 571, 428

142, 857 x 5 = 714, 285

142, 857 x 6 = 857, 142

142, 857 x 7 = 999, 999

The same order of numbers appears in each multiple. The order is the original number 142, 857. Then, the last multiple is 999, 999. The digits in the number 142, 857 add up to 27, a multiple of 9. The digits in the last number 999, 999 also add up, of course since they are all nines, to a multiple of 9, i.e., 54. Also, the two sets of three nines in 999, 999 each add up to 27 which is, again, the sum of the digits in 142, 857.


Extensions of the Problem 

The following worksheet is copied from the website It contains word problems involving "mystery numbers." The answers to these problems are not provided, therefore, they truly are mystery numbers because I do not know the answers or how to arrive at them. However, I included this sheet because I thought it may extend this topic, and that if someone else ever reads this page, she/he may be able to figure out the answers and use this in the classroom.

Mathematics Problem Solving
Volume 6, Number 2, September 18, 2000

Place Value - Mystery Numbers

1. This mystery number has 4 digits. Every digit is an odd number. None of the digits is a 9. Every digit in the number is different. The smallest digit is in the thousands place. The greatest digit is in the ones place. The preceding describes two possible numbers. The mystery number is the greater of those two numbers. What is the mystery number? 

2. This mystery number has 5 digits. There is a 4 in the ten thousands place. None of the other digits is a 4. What is the smallest number that this mystery number can be?  

3. This mystery number has 6 digits. If you add one to this number it will be a 7 digit number. What is the mystery number?  

4. This mystery number is one half of a billion. How many zeros are in this number?

Author & Contact
Pam Joseph