15 January 2016

Puzzle: Inverted License Plates (or why Sharmila does not go for lunch with me anymore :-) )

A couple of days back, after Sharmila had returned from India, we went out for lunch. We were discussing her experiences in India when the topic went to the new restrictions of driving in New Delhi. If you were not aware, New Delhi has promulgated new rules (I am not sure if it is effective still) that stipulates that cars with odd or even number license plates can be driven on alternate days. So, one day, if cars with only even number license plates can be driven, the next day it would be for the cars with odd number license plates. This, if I am not very mistaken, is to keep the pollution under control.

As a side story, I am told that women drivers are exempt from this rule. Which prompted me to suggest to Sharmila that we lost a great business opportunity to sell “burkhas” (the head to toe garb that many Muslim ladies wear) in New Delhi. Who is going to figure out whether it is a guy or a girl driving once they put a “burkha” on πŸ™‚

In any case, another idea came to mind then. How about those guys with license plates that are reversible? Like if I had a license plate “6666”, I can easily take out four screws and put them back on with the license plate looking “9999”. In practice, that is not possible since there are other things written on the license plate that will clearly expose the trick but that did give rise to today’s puzzle:

How many license plates can be there which when reversed (and reattached to the car) gives a legitimate alternate parity (odd becomes even, even becomes odd) license plate?

Assume the following:

  1. License plates are no more than 4 digits long
  2. A license plate cannot start with 0
  3. 0,1,6,8,9 are the digits that while reversed looks like legitimate digits
  4. There cannot be two license plates which look the same on the roads ever

Posted January 15, 2016 by Rajib Roy in category "Puzzles


  1. By Bijit Bose on

    This wont work in Indian metros :-). We use a HSNP here, which are rivetted, not screwed. Every time you open the plate, you need a new one.

  2. By Rajib Roy on

    Bijit, Pratyush and Sanjib, the discussion so far has been errr… “rivetting” πŸ™‚ Now solve the problem πŸ™‚

  3. By Bijit Bose on

    Thanks for your encouragement – I am sure that will push me to know you more Sanjib. Now let us give the control back to Rajib as far as this post is concerned.

  4. By Sanjib Banerjee on

    You won’t believe what I did. My car key is inside the trunk along with my stuffs and I’m 24 miles away to catch my flight to Atlanta in next 75 mins. Waiting for AAA/Avis to give me some solution unless you’ve one!

  5. By Nitin Thakur on

    First digit can have 4 numbers, second digit can have 5 numbers, third digit can have 5 numbers and the last digit can have only 1 number (6 or 9 , because of your last requirement). 4 times 5 times 5 times 1 = 100

  6. By Nitin Thakur on

    Jay, the last digit cant take 0, 1 or 8, since reversing those does not change odd to even. It can only take 6 or 9, and not both because of the last requirement

  7. By Rajib Roy on

    Nitin to get past “reversing” definition, think about a license plate and how you would put it inverted. Write the number on a paper and turn is upside down. Also, you can have license plates of three digits, two digits etc.

  8. By Rajib Roy on

    Jay, are you not going to have duplicates? Remember the rule you cannot have two cars with the same plates on the same day.


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