# Why The Addition Number Trick Works

In the Addition Number Trick, my latest post in the Mathematical Palette (sorry, this site is already gone), I shared about a number trick I learned when I was still in the elementary school. The trick involves (1) letting a friend write a number  and then (2) predicting the sum of the five numbers, including the four numbers (3 – 6) that you are about to write in alternate.  The order of writing is indicated by the circled numbers in the figure below. To get the sum of the five numbers, subtract 2 from the first number and then affix it to the leftmost part (543 becomes 2541).  Next, your friend and you will write the numbers in alternate, and you should make sure that each digit on the same place value add up to 9 (see detailed instructions). After adding the numbers, the sum should be correct.

##### Why The Number Trick Works

The trick works using any number of digits as long as all numbers have the same number of digits. Consequently, the following discussion will be general.

From the figure, we see the following pattern.

1. The ones digit of the sum is 2 less than the ones digit of the first number.
2. The remaining numbers to the left of the ones digit of the sum are the same as those of the first number on the same place value.
3. The number at the leftmost digit is always 2.
##### The Reasons
1. Since each pair of digit on the same place value add up to 9, the ones digit of the four numbers (not including the first number), add up to 18. Now, the sum of 8 (the ones digit of 18) and any number is 2 less than the number (Can you see why?).
2. Since the ones digit is more than 20, we add 2 to the next digit. The next digits always add up to 18, so adding 2 to them, will always equal 20, which gives us 0. Now, 0 added to any digit on the first number is equal to that digit. This means, that all the digits of the first number and the sums are equal if they have the same place value.
3. Since we always add 2 to the next digit on the left, the leftmost digit of the sum is 2.

There is another variation to this number trick, but I will leave it as an exercise.  Will the trick work if

1. the ones digit is less than 1?
2. the ones and the tens digit are less than 1?