### Brain Teasers

# Reversing Numbers

If a number consisting of three different digits is subtracted from its reverse, the answer is the same three digits in yet another order. What is the number?

### Answer

The only possibility is: 954 - 459 = 495.Hide Answer Show Answer

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## Comments

Your solution says 'one possibility'. I think it's the only one. (however the 'negative' also works)

Let the 3 digit number be denoted xyz, that is

100x + 10y + z.

Now reverse and subtract gives

100x + 10y + z - 100z - 10y - x

= 99(x - z).

Hence x-z is a multiple of 99, one of the following

2 198

3 297

4 396

5 495

6 594

7 693

8 792

9 891

Now, as x and z are 2 of the digits in these numbers above, we must be able to subtract one of the three from another of the three to give the multiple (2-9) on the left.

By elimination we see that this works for only:

5 (*99=) 495

because 9-4 = 5

hence x=9, z=4 so y=5

954

459

----

495

The negaitve solution also works.

459

954

----

-495

Let the 3 digit number be denoted xyz, that is

100x + 10y + z.

Now reverse and subtract gives

100x + 10y + z - 100z - 10y - x

= 99(x - z).

Hence x-z is a multiple of 99, one of the following

2 198

3 297

4 396

5 495

6 594

7 693

8 792

9 891

Now, as x and z are 2 of the digits in these numbers above, we must be able to subtract one of the three from another of the three to give the multiple (2-9) on the left.

By elimination we see that this works for only:

5 (*99=) 495

because 9-4 = 5

hence x=9, z=4 so y=5

954

459

----

495

The negaitve solution also works.

459

954

----

-495

^^^^^

wow

wow

Excellent Job, Tony!

A nice puzzle, and a great explanation, Tony!

It was a nice puzzle!

I did it using logic instead of math.

1) Since the middle number stays the same, the number must have a nine in it, otherwise the middle number will subtract itself to become zero, and zero can't be the first or last digit, so it can't be any digit.

2) If the number has a nine in it, then the difference between the other two numbers must be one, otherwise they can't produce a nine when the larger is subtracted from the smaller.

3) Lastly, the two other numbers must total nine, or they can't produce each other when subtracted from nine.

So the digits had to be 4, 5 & 9. The reverse of the number had to be the sum of the other two, which mean the nine had to go last.

I'm sure more logic could have reduced it to a single choice, but at this point it was either 459 or 549, so I tried 459 and it worked.

I did it using logic instead of math.

1) Since the middle number stays the same, the number must have a nine in it, otherwise the middle number will subtract itself to become zero, and zero can't be the first or last digit, so it can't be any digit.

2) If the number has a nine in it, then the difference between the other two numbers must be one, otherwise they can't produce a nine when the larger is subtracted from the smaller.

3) Lastly, the two other numbers must total nine, or they can't produce each other when subtracted from nine.

So the digits had to be 4, 5 & 9. The reverse of the number had to be the sum of the other two, which mean the nine had to go last.

I'm sure more logic could have reduced it to a single choice, but at this point it was either 459 or 549, so I tried 459 and it worked.

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