The Peculiar License Plate

Logic Level 3

When I acquired my race car, the first thing I had to do was to get a license plate. The plate I got had a peculiar number on it.

It consisted of 5 different numbers and by mistake when I fixed it upside down, the number could still be read, but the value had increased by 78633.

What was my actual license plate number?

details and assumptions

  1. 1 upside down is 1
  2. 2 upside down is 2
  3. 5 upside down is 5
  4. 6 upside down is 9
  5. 8 upside down is 8
  6. 9 upside down is 6


The answer is 10968.

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3 solutions

Chew-Seong Cheong
Jul 14, 2014

Let the original license plate number be a b c d e abcde and the reversed one A B C D E ABCDE . We know that the numbers readable in reverse are 0 , 1 , 6 , 8 , 9 0,1,6,8,9 . We are given that A B C D E e d c b a = 78633 ABCDE-edcba=78633 or:

A e 7 B d 8 C c 6 D b 3 E a 3 \begin{matrix} \\ - \\ \end{matrix}\begin{matrix} A \\ e \\ 7 \end{matrix}\begin{matrix} B \\ d \\ 8 \end{matrix}\begin{matrix} C \\ c \\ 6 \end{matrix}\begin{matrix} D \\ b \\ 3 \end{matrix}\begin{matrix} E \\ a \\ 3 \end{matrix}

Let's look first pair of digits A A and e e . For the difference to be 7 7 , there are only two cases, ( A , e ) = ( 9 , 1 ) (A, e)=(9,1) or ( 8 , 1 ) (8,1) . If ( A , e ) = ( 9 , 1 ) (A,e)=(9,1) , then the last pair of digits are ( 1 , 6 ) (1,6) and . . . . 1 . . . . 6 . . . . 3 ....1-....6 \ne ....3 , therefore, ( A , e ) ( 9 , 1 ) (A,e)\ne(9,1) and ( A , e ) = ( 8 , 1 ) (A,e)=(8,1) . And we note that . . . . 1 . . . . 8 = . . . . 3 ....1-....8=....3 .

Similarly, for .(B...-.d...=.8...), ( B , d ) ( 8 , 0 ) (B,d)\ne(8,0) because ( . . . D . . . . b . = . . . 1. (...D.-...b. =...1. instead of . . . 3. ...3. .. Therefore, ( B , d ) = ( 9 , 0 ) (B,d)=(9,0) and ( D , b ) = ( 0 , 6 ) (D, b)=(0,6) , and we note that . . . 0. . . . 6. = . . . 3. ...0.-...6.=...3. .

Now, the remaining ( C , c ) (C,c) . ( C , c ) ( 0 , 0 ) ( 1 , 1 ) ( 8 , 8 ) (C,c)\ne(0,0)\ne(1,1)\ne(8,8) , else . . C . . . . c . . = . . 9.. ..C..-..c..=..9.. instead of . . 6.. ..6.. . ( C , c ) = ( 6 , 9 ) (C,c)=(6,9) fits perfectly.

Therefore, the answer is 89601 10968 = 78633 89601-10968=78633 , and the original plate number is 10968 \boxed{10968} .

Actually depends on how you write the digit, 2 and 5 could also reversed. However the answer is not change

Tran Hieu - 5 years, 5 months ago

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Although 2 and 5 are mirror images of each other, when turn upside down (turn 18 0 180^\circ ), 2 is 2 and 5 is 5. The digits are already stuck on the license plate, if we flip it we will see the back. We can only turn it by 18 0 180^\circ .

Chew-Seong Cheong - 5 years, 5 months ago

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Yes, I mean when reversed 2 is 2 and 5 is 5, but that won't change the outcome

Tran Hieu - 5 years, 5 months ago

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@Tran Hieu I see. But if the numbers are not digital seven-segment format. They don't look alike went rotated.

Chew-Seong Cheong - 5 years, 5 months ago

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@Chew-Seong Cheong Well I figure if the 1 could be rotated then the number should be in that format. Otherwise the 1 when rotated will look like a fishing hook

Tran Hieu - 5 years, 5 months ago

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@Tran Hieu You are right.

Chew-Seong Cheong - 5 years, 5 months ago
Henny Lim
Jan 12, 2016

Readable numbers in reverse positions: 0, 1, 6, 8, and 9

Increase of 70k something --> must be 8 and 1 (difference of 7)

So I put it: 8xxx1 - 1xxx8 = 78633

Then I looked on the 10th value. The difference is 3, but remember we have taken 10 previously to make 1(0) minus 8 becomes 3. So the difference should be 4. It must be 0 and 6.

So I put it: 86x01 - 10x68 = 78633

Now there are only one pair of numbers left --> with difference of 6. Since we have taken 1 from 10th value to make zero minus six becomes 3 as well; the only possible combination is 6 and 9.

Therefore: 86901 - 10968 = 78633

We can do a check and find the equation is correct.

Naimish Khara
Jul 13, 2014

10968

+78633

89601

Which is seen when it is up side down....

This is just the answer. This is not a solution..

Shabarish Ch - 6 years, 11 months ago

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LOL. You are right. I solved it in a very simple manner.1st digit and last digit should be 1 and 8 respectively as the difference is more than 70000. And adding 78633 Gave the only possible answer.

Kushagra Sahni - 6 years, 11 months ago

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I did it in a somewhat same manner too..

Shabarish Ch - 6 years, 11 months ago

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