# Solution

The digits, which can be read upside down are 0,1,6,8 and 9. Thus, the number comprised only these digits. Suppose the number was a four-digit one, say 6890, then turned upside down it will appear as 0689.

Now, the upside-down number is greater than the actual one by 78633. Suppose the actual number is "x" and the upside down number is "y", then y - x = 78633. So, y = 78633 + x. This means that the upside down number is greater than 78633. So, "y" has to be, at least, 78634 and since "y" is a five-digit number, it can be, at most, 99999. Therefore, the digit in the ten thousand's place can be 7, 8, or 9. But, "7" is not among the digits which can be read upside down; therefore, the digit in the ten thousand's place in "y" can be either 8 or 9.

The digit in the unit's place in the actual number "x" will appear as the digit in the ten thousand's place in the upside down number "y". Does that sound rubbish? Okay, let's consider an example. Let's say our actual number is 88916; when viewed upside down, this will appear as 91688. Write down the number on a piece of paper and, then, turn the paper upside down. So, the "8" in the ten thousand's place in the actual number is the digit in the unit's place in the upside down number, and vice versa. Also, notice that "9" appears as "6" and "6" appears as "9".

So, if the digit in ten thousands place in "y" is 8 or 9, then the digit in the unit's place in "x" has to be either "8" or "6".

"y" is greater than "x" by 78633. Consider the last digit, "3". If the last digit in "x" is "8", then the last digit in "y" has to be "1" because only then we will have 1 - 8 = 3 (this will be 11 - 8 after the carry). If, on the other hand, the last digit of "x" is "6" then last digit of "y" has to be "9" because only then we will have 9 - 6 = 3.

So, we have seen that the first digit of "y" can be either 8 or 9 and it's last digit can be either 1 or 9. So, the (first digit, last digit) pairs possible for "y" are (8, 1); (8, 9); (9, 1); (9, 9). Now, the registration number has distinct digits; so, we can rule out the last pair. Only three (first digit, last digit) pairs are possible: (8, 1); (8, 9); (9, 1).

Let's consider the pair (8, 1): The other three digits which can make up the five-digit number are 0, 6 and 9 (all the digits are distinct and no digit is repeated). Now, 0, 6 and 9 can be arranged in six ways: 069, 096, 609, 690, 906, and 960. So, the five-digit number could be: 80691, 80961, 86091, 86901, 89061, and 89601. If these numbers are turned upside down, we will get "x", which will be: 16908, 19608, 16098, 10698, 19068, and 10968. Let's find the difference: (80691 - 16908), (80961 - 19608), (86091 - 16098), (86901 - 10698), (89061 - 19068), and (89601 - 10968). Voila! Indeed, 89601 - 10968 = 78633. So, we have found our number; y = 89601 and x = 10968. Now, x is the actual number. The actual number is 10968.

Shall we continue to check whether other numbers are possible? It is not likely, but let's do it.

Now, we will consider the pair (8, 9) - the first and last digits in "y". The other three digits which can be squeezed in between are 1,6, and 0. The three digits (0, 1, 6) can have six different arrangements among them: 106, 160, 016, 061, 610 and 601. So the five-digit number after considering these combinations could be: 81069, 81609, 80169, 80619, 86109 and 86019. The corresponding upside down numbers will be: 69018, 60918, 69108, 61908, 60198, and 61098. Just one glance at these numbers is enough to tell us that none of the pairs will have a high difference of 78633.

Let's consider the last pair (9, 1): The three other digits that can be squeezed within are 6, 8 and 0. These three digits can be arranged in six different ways: (680), (608), (860), (806), (068), and (086). The five-digit number would be: (96801), (96081), (98601), (98061), (90681), and (90861). The upside down numbers would be: 10896, 18096, 10986, 19086, 18906, and 19806. Let's work out (96801 - 10896), (96081 - 18096), (98601 - 10986), (98061 - 19086), (90681 - 18906), and (90861 - 19806). Thankfully, none of the differences work out to 78633.

Phew! That was hard work. I am sure there must be a much easier way to solve this. Besides, I have a feeling I must have messed up while writing "9" as "6" and "6" as "9". Please do let me know how you can solve this within minutes.

• Union Public Service Commission - www.upsc.gov.in
• IIT-Kharagpur - www.iitkgp.ac.in
• Indian Statistical Institute - www.isical.ac.in
• Indian Institute of Technology Madras - www.iitm.ac.in
• Indian Institute of Management, Ahmedabad - www.iimahd.ernet.in
• Indian Institute of Mass Commission - www.iimc.nic.in
• IIT Bombay - www.iitb.ac.in