Determining a hidden digit
Think of a positive integer, call it X. Shuffle the decimal digits of X, call the resulting number Y. Subtract the smaller of X,Y from the larger, call the difference D. D has the following property: Any non-zero decimal digit of D can be determined from the remaining digits. That is, if you ask someone to hide any one of the non-zero digits in the decimal representation of D, then you can try to impress the other person by figuring out the hidden digit from the remaining digits. How is this done? Why does it work?
The short short version is that this is an artifact of the idea of casting out nines: the sum of a number’s digits gives the same remainder on division by 9 that the original number does. For instance, 431=9×47+8, and 4+3+1=8. Since X and Y have the same digits, they have the same digit-sum, so they have the same remainder on dividing by 9. But this means that their difference D is a multiple of 9; and that means that D’s digit-sum is a multiple of 9 (since it must give the same remainder on dividing by 9 that D does). When a non-zero digit is hidden, it’ll reduce the digit-sum of D by that much, so all you have to compute is the digit from 1—9 (and note that there will always be a unique digit – this is why the ‘nonzero’ restriction is there, because otherwise there’d be no way to tell a zero from a nine) to add to the digit-sum of D to bring it up to a multiple of 9.
If you’re interested in the concept, type in the phrase “casting out nines” in your search engine of choice.
In the example you gave, the basic idea is that you know that X and Y have the same remainder when divided by 9 since they share the same digits1. Hence, the difference must be a multiple of 9. Since we know the sum of the digits of any multiple of 9 is itself a multiple of 9, you can deduce the last digit.
(In the case that the sum is itself a multiple of 9, you can use the fact that they hid a non-zero digit to deduce that the remaining digit is in fact a 9.)
Your Answer
More puzzles to try-
What is the logic behind these ?
3 + 3 = 3 5 + 4 = 4 1 + 0 = 3 2 + 3 = 4 ...Read More »Defective stack of coins puzzle
There are 10 stacks of 10 coins each. Each coin weights 10 gms. However, one stack of coins is defective ...Read More »Which clock works best?
Which clock works best? The one that loses a minute a day or the one that doesn’t work at all?Read More »(Advanced) Cheryl’s Birthday Puzzle
Paul, Sam and Dean are assigned the task of figuring out two numbers. They get the following information: Both numbers ...Read More »Five greedy pirates and gold coin distribution Puzzle
Five puzzleFry ship’s pirates have obtained 100 gold coins and have to divide up the loot. The pirates are all ...Read More »Magical flowers!!
A devotee goes to three temples, temple1, temple2 and temple3 one after the other. In front of each temple, there ...Read More »Tuesday, Thursday what are other two days staring with T?
Four days are there which start with the letter ‘T‘. I can remember only two of them as “Tuesday , Thursday”. ...Read More »How could only 3 apples left
Two fathers took their sons to a fruit stall. Each man and son bought an apple, But when they returned ...Read More »How Many Eggs ?
A farmer is taking her eggs to the market in a cart, but she hits a pothole, which knocks over ...Read More »HARD MATHS – How much faster is one train from other Puzzle
Two trains starting at same time, one from Bangalore to Mysore and other in opposite direction arrive at their destination ...Read More »Most Analytical GOOGLE INTERVIEW Question Revealed
Let it be simple and as direct as possible. Interviewer : Tell me how much time (in days) and money would ...Read More »Lateral thinking sequence Puzzle
Solve this logic sequence puzzle by the correct digit- 8080 = 6 1357 = 0 2022 = 1 1999 = ...Read More »How did he know?
A man leaves his house in the morning to go to office and kisses his wife. In the evening on ...Read More »Pizza Cost Math Brain Teaser
Jasmine, Thibault, and Noah were having a night out and decided to order a pizza for $10. It turned out ...Read More »Which letter replaces the question mark
Which letter replaces the question markRead More »Which room is safest puzzle
A murderer is condemned to death. He has to choose between three rooms. The first is full of raging fires, ...Read More »Richie’s Number System
Richie established a very strange number system. According to her claim for different combination of 0 and 2 you will ...Read More »Srabon wanted to pass
The result of math class test came out. Fariha’s mark was an even number. Srabon got a prime!! Nabila got ...Read More »Become Normal!!
Robi is a very serious student. On the first day of this year his seriousness for study was 1 hour. ...Read More »Sakib Knows The Number!
Ragib: I got digits of a 2 digit number Sakib: Is it an odd? Ragib: Yes. Moreover, the sum of ...Read More »