I am always intrigued by good number patterns; here's one that I've used with my students which requires only basic arithmetic. For reasons that I'll explain below, I'll call this "Kaprekar's game".

1. Choose any four digit number where not all the digits are the same (not divisible by 1111). (The number can start with a 0.)
2. Rearrange the digits from largest to smallest and then from smallest to largest.
3. Subtract the smaller rearrangement from the larger.
4. Take your answer from step 3 and go back to step 2.

For example, if you start with 4997, this is what happens:

Now ask your students to try to figure out the maximum number of steps you can play the game before repeating a number. The amazing answer is that if you play Kaprekar's game on any four digit number (other than multiples of 1111) you will reach 6174 in at most seven steps!

I first saw this property of 6174 as an elementary problem in the American Mathematical Monthly [1]. The original source was a short note in an obscure journal by D. R. Kaprekar of Devlali, India [2]; since that time 6174 has been referred to as Kaprekar's constant.

Of course there's no need to stop with four digits. What happens if you perform this operation with a two, three, or five digit number? Is there always a fixed point? Can there be more than one fixed point (not counting 0, which is always a fixed point)? Such questions are the basis for the concepts of iteration and dynamical systems, and can be introduced to students with very little background.

References

1. Elementary problem E2222, American Mathematical Monthly, March 1970, p. 307.
2. D. R. Kaprekar, "Another Solitaire Game", Scripta Mathematica, September 1949, p. 244-5.