I have a number machine. I say x gives y if when x goes in, y comes out. For any numbers x and y, by "xy" I mean x followed by y. Here are my two rules.

1x gives x.
For example, 13 gives 3.
If x gives y, then 2x gives yy.
For example, 213 gives 33 since 13 gives 3.

What is a number that gives itself?

(In reply to Solution by Jun)

I thought about that... but the number machine only seems to work on numbers beginning with 1_ or 21_

2x gives yy... so x = 2 and y = 2. But 2 doesn't seem to work.

Since 10^?+x will never equal x, we must be looking at the last sequence.

So... 2 __ = __ __

But putting the first sequence into the second one, you get

2(1 __)=(__)(__)

So 21 followed by a number equals the same number twice.

Since 121 gives 21, 2121 gives 2121.

This number must be 21, so 121 gives itself.

Posted by Gamer on 2003-06-08 04:48:26