Here's a trick I like that looks somewhat similar to the previous examples, yet whose explanation may elude some, and which at any rate connects to interesting deeper math. Alas, its one fault is that most people aren't familiar with how to carry out the required large exponentiations.

But, here, I'll link you to a calculator you can use for this purpose.
Pick any three-digit even number you like. Write a 1 after it, and then write in an exponent of 1 followed by three 0s. (For example, if your original number was 712, it would become 7121

^{1000})

Carry out this exponentiation. The result will have lots of digits; in particular, it will end in three 0s followed by a 1. Get rid of those. The result will still be huge; cut it down to size by keeping only the last three digits at this point. (For example, 7121

^{1000} is very large, and its last digits are "...2204035920001". Getting rid of the 0001 chunk and then keeping only the last three digits, we end up with 592). This value will be even, so halve it, and then tell me what you get. (In this case, that would be 592/2 = 296).

At this point, armed with nothing else, I will readily recover your original number. In fact, the most surprising thing is the

*way* in which I will recover your original number: I raise the "magic number" 8221 to what you gave me, chop off the last digit, and then keep only the last three digits. [E.g., 8221

^{296} has last digits "...733817121". Chopping off the last 1 and then keeping only the last three three digits, I get "712" back, just as you started with].

The challenge to you (after you are able to actually try out enough examples to assure yourself this does indeed work) is to explain:

*why* does this work?

(In particular, you should know I can do this trick for

*any* number of digits, in

*any* base, just changing the "magic number" accordingly.)