Cracking the Coding Interview, 4 Edition - 150 Programming Interview Questions and Solutions

Solutions to Chapter 20 | Hard
20.1 Write a function that adds two numbers. You should not use + or any arithmetic operators.
SOLUTION
pg 91
To investigate this problem, let’s start off by gaining a deeper understanding of how we add
numbers. We’ll work in Base 10 so that it’s easier to see. To add 759 + 674, I would usually add
digit[0] from each number, carry the one, add digit[1] from each number, carry the one, etc.
You could take the same approach in binary: add each digit, and carry the one as necessary.
Can we make this a little easier? Yes! Imagine I decided to split apart the “addition” and
“carry” steps. That is, I do the following:
1. Add 759 + 674, but “forget” to carry. I then get 323.
2. Add 759 + 674 but only do the carrying, rather than the addition of each digit. I then
get 1110.
3. Add the result of the first two operations (recursively, using the same process described
in step 1 and 2): 1110 + 323 = 1433.
Now, how would we do this in binary?
1. If I add two binary numbers together but forget to carry, bit[i] will be 0 if bit[i] in a and
b are both 0 or both 1. This is an XOR.
2. If I add two numbers together but only carry, I will have a 1 in bit[i] if bit[i-1] in a and b
are both 1’s. This is an AND, shifted.
3. Now, recurse until there’s nothing to carry.
1 int add_no_arithm(int a, int b) {
2 if (b == 0) return a;
3 int sum = a ^ b; // add without carrying
4 int carry = (a & b)