Taken from CodingBat
Factorial source
Given n of 1 or more, return the factorial of n, which is n * (n-1) * (n-2) ... 1. Compute the result recursively (without loops).
Bunny Ears source
We have a number of bunnies and each bunny has two big floppy ears. We want to compute the total number of ears across all the bunnies recursively (without loops or multiplication).
Example:
bunnyEars(0) → 0
bunnyEars(1) → 2
bunnyEars(2) → 4
Bunny Ears 2 source
We have bunnies standing in a line, numbered 1, 2, ... The odd bunnies (1, 3, ..) have the normal 2 ears. The even bunnies (2, 4, ..) we'll say have 3 ears, because they each have a raised foot. Recursively return the number of "ears" in the bunny line 1, 2, ... n (without loops or multiplication).
Example:
bunnyEars2(0) → 0
bunnyEars2(1) → 2
bunnyEars2(2) → 5
Triangle source
We have triangle made of blocks. The topmost row has 1 block, the next row down has 2 blocks, the next row has 3 blocks, and so on. Compute recursively (no loops or multiplication) the total number of blocks in such a triangle with the given number of rows.
Example:
triangle(0) → 0
triangle(1) → 1
triangle(2) → 3
Sum Digits source
Given a non-negative int n, return the sum of its digits recursively (no loops). Note that mod (%) by 10 yields the rightmost digit (126 % 10 is 6), while divide (/) by 10 removes the rightmost digit (126 / 10 is 12).
Example:
sumDigits(126) → 9
sumDigits(49) → 13
sumDigits(12) → 3
You can click through to the CodingBat website for a whole bunch of exercises. These exercises are in Java, but you can do them in any language you want.