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# Subtraction Surprise

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Elizabeth and Serena from Withington Girl’s School in the UK described what happens in the video:

In the examples in the video, you subtract the number you think of by the reversed order of your number. Afterwards, you use the answer to add to the reversed order of the answer and get 1089, if the first and the last digits are different. The number that you start with [can be] different.

Surya, Na'ima and Srinika, Abdulla and Issa from British School Al Khubairat in the UAE tried out some more numbers. Surya wrote:

I have done a few calculations and all the answers add up to 1089, for example,

875 - 578 = 297 + 792 = 1089 and 863 - 368 = 495 + 594 = 1089.

Issa added:

You cannot use a number with the same first and 3rd digit. This is because reversed it will equal the same (101 reversed is 101).

Anirudh, Ishbel, Abigail, Ethan, Gemma, Zaina, Ishaan and Luke from Cambourne Village College made some interesting observations about what happens during the process. Here is Abigail's work (click on the image to open a larger version):

Ishaan from Cambourne Village College used an example to show why this leads to a final answer of 1089:

Anh (Alex) from British Vietnamese International School in Vietnam used algebra and some systematic trials to show why the answer is always 1089 (click on the image to open a larger version):

Ci Hui Minh Ngoc Ong from Kelvin Grove State College Brisbane in Australia used similar notation, but showed exactly how the column subtraction and addition works algebraically, in particular borrowing and carrying (click on the image to open a larger version):

Alina, Gemma and Ishbel from Cambourne Village College wondered what would happen for 2 and 4 digit numbers. Here is Alina's work (click to enlarge):

Elizabeth and Serena also experimented with negative numbers. They wrote:

All the examples in the video worked out, however they didn’t use any negatives or use any 0 digits in the 3 digit number. No matter if you use a zero in the 3 digit number, make the first digit smaller than the last digit (which they didn't do in the video) or use a negative 3 digit number, the answer always comes to 1089. For example, if you use the number -293, the reverse would be -392.
-293 - -392 comes to -293 +392 which equals 099, and 099 + 990 is 1089. In addition, we noticed that with the examples with 0 as a digit in the number, the first subtraction

usually came to 099 as well.

Choose two digits and arrange them to make two double-digit numbers. Now add your double-digit numbers. Now add your single digit numbers. Divide your double-digit answer by your single-digit answer. Try lots of examples. What happens? Can you explain it?

What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?