Is there sufficient information?
An 8-digit number with no 0's (ORIGINAL) has its digits scrambled to form a smaller number (REARRANGEMENT):
ORIGINAL − REARRANGEMENT = FINAL .
If FINAL has no 0's and contains 8 distinct digits, then which of the digits from 1 to 9 is missing from FINAL?
If there is insufficient information to determine, enter your answer as 0.
The answer is 9.
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2 solutions
Great question!
You have shown that 9 is necessary. How can we show that 9 is sufficient by finding an example? It is not immediately clear to me that there exists (say) "final - original = 12345678".
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By programming, I found at least one instance 22351000 ( 2 2 3 5 1 0 0 0 − 1 0 0 0 5 3 2 2 = 1 2 3 4 5 6 7 8 ). And probably many many more possible.
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ORIGINAL isn't supposed to have any 0's, but we can modify your solution as follows:
33462111 - 21116433 = 12345678
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@Denton Young – Whoops, didn't see the condition. Yes, just increment each digit by 1.
As a further question, is there an ORIGINAL whose digits are all different?
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@Ivan Koswara – That is a very good question. Someone who has access to a Python-programmable computer will have to answer it. Unless they want to search every possibility by hand. And remember that FINAL doesn't have to be 12345678, the digits can be in any order. FINAL being 24318675, for example, would be just fine.
I don't actually know of an example offhand, which is why I used the phrasing of "IF (listing conditions for FINAL).". I can try and find one...
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If we don't want to prove the existence of such a number, why don't we just add a line of "Suppose that we can assume that there is such an integer ORIGINAL that satisfy all these conditions." ?
let original = 1 0 7 x 1 + 1 0 6 x 2 + . . . . . . . . . . . . . . . . . + 1 0 1 x 7 + x 8
where x 1 , x 2 . . . . . . . . . . x 8 are digits
final = original - some random rearrangement of original
now original = ( ( 1 0 7 − 1 ) x 1 + ( 1 0 6 − 1 ) x 2 + . . . . . . . . . . . . . . . . . + ( 1 0 1 − 1 ) x 7 ) + ( x 1 + x 2 + . . . . . . . . . . . . . . . . x 7 + x 8 )
= 9 a + ( x 1 + x 2 + . . . . . . . . . . . . . . . . x 7 + x 8 )
similarly any rearrangement = 9 b + ( x 1 + x 2 + . . . . . . . . . . . . . . . . x 7 + x 8 )
thus final = original - some random rearrangement of original
=( 9 a + ( x 1 + x 2 + . . . . . . . . . . . . . . . . x 7 + x 8 ) − ( 9 b + ( x 1 + x 2 + . . . . . . . . . . . . . . . . x 7 + x 8 ) ) = 9 ( a − b )
thus final is a multiple of 9
therefore sum of 8 digits must be a multiple of 9
( 1 , 8 ) , ( 2 , 7 ) , ( 3 , 6 ) , ( 4 , 5 ) all add upto 9 , thus the missing digit has to be 9
FINAL has to have a digital root of 9, because ORIGINAL and REARRANGEMENT have the same digital root, being composed of the same numbers.
1 + 2 + ... + 9 = 45, which has a digital root of 9. One of those digits is missing, and the remaining 8 have to sum to a digital root of 9. This is only possible if 9 is removed from the set.
We show that this can be achieved: 33462111 - 21116433 = 12345678