Expert: Josh - 10/19/2008

Question
Problem: The License plate on my car contains 5 different digits. My son installed it upside down, yet it still shows a 5 digit numeral. The only thing is the new number exceeds the original number by 63, 783.  What was the original number on the license plate?
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I tried to do this problem and no luck. I know the number should consists on 1, 6,8,9 because only those look the same upside down. I tried to work between the numbers subtracting them to get 63,783 but then the numbers all have to be different in a numeral because at the beginning it said "different digits" therefore nothing worked. Please help me I have this question for Tuesday and no idea how to start. Can you please explain to me how to do it not just give an answer. I would appreciate it. THANK YOU


Answer
Hi Anna,

Your observations are pretty much spot on. Let me add one thing. The digit 0 should also be included amongst the list {1,6,8,9}.

Since all digits are different, why don't we write the 5 digit string as "abcde" where a,b,c,d and e denote the unknown digits.

In the metric system, numbers are represented in base-10. Therefore, the literal "abcde" has value N=a*10000+b*1000+c*100+d*10+e.

Let's suppose we get the string "fghij" when the plate is turned up-side-down. Again, represent this as R=f*10000+g*1000+h*100+i*10+j.

We know for a fact that R-N=10000(f-a)+1000(g-b)+100(h-c)+10(i-d)+(j-e)=63783

...I am beginning to wonder if this is a trick question which is not meant to be solved mathematically.

Let's look at the most significant digit on the left. If R-N=63783,  it stands to reason that R="fghij" must begin with a digit no smaller than 6. The digit f must be one of {6,8 or 9}.

Neither 8, nor 9 can produce "6" when one of the digits in {0,1,6} is subtracted from it. Thus, we conclude that "6" and "0" is the only viable option for the digit "f" in R and digit "a" in N. In particular, "f" must be greater than "a", so R="6ghij" and N="0bcde".

If I have understood the question correctly, "e" is a 180 degree rotated version of "f". So, "e"=9. By symmetry, "j"=0 (after the flip). At this stage, we have R="6ghi0" and N="0bcd9".

We have {0,1,8} left. The more I think about this, the more ambiguous the question seems to be. e.g., how are the digits actually read -- it seems obvious, but it is not clearly stated. The following seems to yield a contradiction. Now, the obvious way to get a difference of "7" is to have h=8, c=1. But "h" should be a 180 degree rotated version of digit "c". So, this is impossible. The remaining possibility is to have something like 9-1, with a borrow from the digit on the right hand side.

...Okay, I've done some work on paper, here is the solution that I've found after retracing the steps.

The question is really tricky. I'll simply state the following to stay on the right track. The previous observations regarding the most significant digits "f" and "a" still hold. Except, we haven't consider the possibility that "f" has been borrowed by its right digit "g". In this case, we can have "f"=8 and "a"=1. Normally, the difference is 2 of course, but with the "borrow" from digit "e", the difference becomes (8-1)-1=6. So, we have R="8ghi1" and N="1bcd8" if our assumptions are right.

This leaves us with {0,6,9} to play with. Again, it's not possible to obtain a middle digit of 7 without any borrowing. We remember (from years of arithmetic drill exercise) that 16-9=7. Focusing on the "ghi" part in R and the "bcd" part in N, if we can get "g" to be smaller than "b", then the digit "g" must borrow 1 from "f", as required by our earlier assumption. So, we should contemplate having "0" on the left hand side of "h" and a "9" on the right hand side of "h". We also need to ensure that "ij"-"de" equals 83.

Trial solution:
R="fghij"="80691" and N="abcde"="16908".

We see that indeed, "ij"-"de"=91-08=83.
The middle is, with borrowing, "16"-"09"=7 by construction.
Now, with 1 subtracted from "g"=0, "g" effectively becomes a "9". Subtracting "b"=6 from it, we get a difference of "3". This plays out the scenario we envisaged in the beginning, (f-1)-a=6.

It's difficult, isn't it? Who would do such a thing, setting such a hard question. I just lost some brain cells...