Subtraction Surprise
Subtraction Surprise printable sheet
In the video below, Alison chooses some three-digit numbers and carries out some calculations which lead to a surprising result!
Watch the video. What do you notice?
Can you figure out the steps that Alison carries out in each calculation?
This video has no sound.
If you can't see the video, click below to see the steps and the examples Alison tried.
Pick any three digit number.
Reverse the digits, so write your number back to front.
Subtract the smaller of your two numbers from the larger one.
Now reverse the digits of the answer you get.
Add the answer to its reverse.
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Choose some three-digit numbers of your own.
(Make sure the first and third digits are different)
Is there a pattern to all the answers?
Now watch the videos again. This time, all three subtractions are carried out at the same time.
You may wish to pause the video at certain points, or watch it several times.
What is the same in each example?
What is different?
Does every example lead to the same answer?
Can you use what you noticed in the video to prove it?
You may also be interested in the other problems in our Tales of the Unexpected Feature.
What do you notice?
”‹What is the same? What is different?
What do you notice?
”‹What is the same? What is different?
What do you notice?
”‹What is the same? What is different?
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.
Why do this problem?
This problem tests students' understanding of place value and provides an opportunity for them to practise calculation methods in a context which leads to a surprising result! We hope that students will be curious about the unexpected outcome and wish to explore, and explain why it happens.
Possible approach
"Choose a three-digit number where the first digit is bigger than the third digit. Reverse the digits, and then subtract your second number from your first one."
Give students a moment to work out the difference.
"Take your three-digit answer and reverse the digits. Add these two numbers together. If your answer wasn't a three-digit number, put a leading zero in the hundreds column before you reverse it."
Once students have had a chance to work out the total, invite a few of them to share their answer. Enjoy the moment of surprise when everyone realises they ALL got the answer 1089.
To investigate why this happens, you may wish to show students the video from the problem.
You might like to invite them to comment on what is the same and what is different in each example. Alternatively, you could invite three students to come out to the board, and present their examples simultaneously, recreating what happens in the video.
In the video, Alison used column subtraction for all three calculations, even though there was a quicker mental method for the third one. This could provoke a useful discussion about choice of methods!
Students might use a variety of methods to perform the subtraction, and create a generalisation based on whichever method they choose (see Possible Support below for one suggestion).
If they use a column subtraction method and they are confident with representing numbers with letters, you could encourage them to use an approach like the one shown in the image below:
Key questions
What is the same in each calculation?
What is different?
Why do you always get a 9 in the tens column when you perform the subtraction?
Why do the hundreds and units of the subtraction always sum to 9?
Possible support
Students can explain why the solution is 1089 without using column subtraction. Here is one possible explanation:
723-327 is the same as 703-307.
This is the same as 700+3-(300+7) = 700-300 +3-7
This gives 400-4, which is 4 times 99.
For any three-digit number, we will follow a similar procedure which will lead to a multiple of 99. There are only 10 of these to check, and they all give 1089.
Possible extension
If you take the four-digit number 4321, reverse the digits, subtract, reverse the digits of the answer and add, you get 10890.
Invite students to explore whether it will work for all four-digit numbers.
If so, can they prove it? If not, can they find the conditions required to give an answer of 10890?