One of Einstein's Favorite Puzzles
It is said that Albert Einstein used to take great delight in baffling friends with the puzzle below.
First, write the number 1089 on a piece of paper, fold it, and hand it to a friend for safekeeping.
What you wrote down is not to be read until you have completed your amazing mental feat.
Next, ask your friend to write down any three-digit number, emphasizing that the first and last
digits must differ by at least two. Close your eyes or turn your back while this is being done.
Better still, have someone blindfold you.
After your friend has written down the three-digit number, ask him to reverse it,
then subtract the smaller from the larger.
Example: 654 - 456 = 198.
Once this is done, tell your friend to reverse the new number.
Example: 198 becomes 891.
Next ask your friend to add the new number and its reverse together.
Example: 198 + 891 = 1089.
If all goes as planned, your friend will be amazed. The number you wrote down
at the start -- 1089 -- will always be the same as the end result of this
mathematical trick.
Here is an explanation of the trick:
Let abc = the original number
where a > c.
Then 100a + 10b + c represents the number.
Then the reverse is cba which is expanded to 100c + 10b + a.
So, abc - cba = (100a + 10b + c) - (100c + 10b + a) = 99a - 99c = 99(a - c).
Since the first and last digits in abc differ by at least 2, then (a - c) is either 2, 3, 4, 5, 6, 7, 8, or 9.
So, 99(a - c) is one of the following: 198, 297, 396, 495, 594, 693, 792, or 891.
The final stage is to add this number to its reverse.
Just repeat what we did before and apply it to this number.
Let's call the number xyz which is 100x + 10y +z.
We want to add xyz to zyx, its reverse.
Looking at the possible numbers for xyz above, we notice that the middle number y is always 9.
And we also notice that the first and third digits always add up to 9, i.e., x + z = 9.
So, xyz + zyx =
100x + 10 y + z + 100z + 10y + x =
100(x + z) + 20y + x + z =
(100 x 9) + (20 x 9) + 9 =
900 + 180 + 9 = 1089.