Mystery Math

Background:

Before computers were common, accountants would use digital roots to quickly check their math. Today, your students can use digital roots to play a trick on their friends while working out math problems in their heads!

Objectives:

In this activity students will:

- learn how to find a number's digital root

- follow a series of steps to solve a problem

- use addition and subtraction to calculate a student's secret number

Pre-Activity Preparations:

Finding a Number's Digital Root

1. Add all of the digits of a number together. For example, 58174 becomes 5 + 8 + 7 + 1 + 4 = 25.

2. If the sum has more than one digit, add all of the sum's digits together. For example, 5874 becomes 5 + 8 + 7 + 1 + 4 = 25 which becomes 2 + 5 = 7.

3. Repeat this process with each subsequent sum until the answer contains only a single digit. This single digit is the original number's digital root.

Notes:

- This trick will work with any size number, not just those with five digits.

Detailed Directions:

1. Have each student write down a five digit number. For example, 81298.

2. Have each student write down a second five digit number by rearranging the order of the digits of their first number. For example, 21889.

3. Have each student subtract their smaller five digit number from their larger five digit number. For example, 81298 - 21889 = 59409.

4. The class doesn't know it, but the digital root of everyone's answer is 9. You will use this fact to do the trick.

5. Have every student cross out one of the digits from their answer. For example, 59409.

(This trick is much easier to do if they do not cross out a zero. The reason will become clear once you know how to do the trick.)

6. Have a student tell you the numbers that are left in their answer. For example, 5, 9, 0 and 9.

7. Calculate the digital root of the numbers they tell you. For example, 5 + 9 + 0 + 9 = 23 which becomes 2 + 3 = 5.

8. You know the digital root of their complete answer is 9 and you know the digital root of the numbers they gave to you is 5, so you know they must have crossed out a 4.

If the student had crossed out a 9, they would have told you 5, 4, 0 and 9. The digital root of those numbers is 9. This tells you that they either crossed out a zero (9 + 0 = 9) or a nine (9 + 9 = 18 which becomes 1 + 8 = 9). Since it is not possible to distinguish a missing zero from a missing nine, you have to prevent the students from crossing out one of them. It has been our experience that saying something like "Cross out any number you like from your answer, except for zero, because zeros are too easy" goes over very well.