Question

Choose any three-digit number, the only constraint is that the first digit within it has to be bigger than the third. (Eg, 321). Reverse the order of the digits, and subtract this from the original number. (Eg, 321-123=198). Reverse the digits of this new number, and add the result of that, to the new number (Eg, 198+891=1089). You’ll notice that the answer has been written on the whiteboard behind you all along the interview. Why is it always 1089?

Answer 1

Hint: Now, 1089 is a magic number since it is the only four-digit numbers that divide their reverse. In this question we need to use mathematical reasoning and logic and understanding of the decimal system to get the answer.

Complete step by step answer:
Now, the number on the whiteboard is 1089.
Let XYZ be the three-digit number such that X>Z then as per the decimal system we can place the digits as:

Hundreds	Tens	Units
X	Y	Z

Now on reversing the number the new number will become ZYX. So, now Z has moved to the hundreds place and X has moved to the units place.
Now the difference between these two numbers will be XYZ-ZXY:
Now to find the difference, the units place will have to gain 9 at tens place and hence the tens place will reduce the digit on the hundreds place by 1, such that the value of hundreds decreases by one hundred.
So the table will now transform to:

Hundreds	Tens	Units
X - 1	Y +9	Z +10

Now on adding this difference to the reverse of the difference the answer will be 1089 always because, overall we will always be taking away one hundred and adding nine tens and ten ones.
For example, let us consider a three-digit number:
762. So here the digit 7 which is at the hundreds place is greater than the digit 2 at the units place. So the first condition Is satisfied.
Next reversing the number, we get 267.
Now we need to find their difference so computing the difference we will get:

Hundreds	Tens	Units
7-1	6+9	2+10
-2	6	7

That is, we will get:

Hundreds	Tens	Units
6	15	12
-2	6	7
4	9	5

So, the difference is 495. Now reversing this difference, we will get the number as 594.
Now, adding 495 and 594:

Hundreds	Tens	Units
4	9	5
+5	9	4
10	8	9

Therefore, the answer stays 1089. Hence irrespective of which three-digit number we choose, as long as the first digit is greater than the third digit, their difference when reversed and added to the difference will always give us 1089, since we will always take away one hundreds and add nine tens and 10ones while finding the difference between the original number and the reversed number.

Note: The first digit should be always greater than the third digit for a given three-digit number, only then the number obtained in the end will be 1089.
The first digit should be always greater than the third digit for a given three-digit number, only then the number obtained in the end will be 1089. We can try this with any other number as well to check that the number always stays 1089.
For example, let the number 541, then its reverse will be 145, and the difference between the original number and its reverse will be 396. The reverse of the difference will be 693. So, the sum of the reverse and the difference will come out to be $396 + 693 = 1089$ .
Hence as long as the digit in hundreds place is greater than the digit in units place, the logic will always hold and the answer will stay as 1089 only.