A Simple Interview Question
I ran into an interview question on leetcode about a number conversion that marked as easy. It reveals the beauty of our numbering notation.
Question
Excel Sheet Column Title
Given a positive integer, return its corresponding column title as appear in an Excel sheet.
For example:
1 > A
2 > B
3 > C
...
26 > Z
27 > AA
28 > AB
...
Example 1:
Input: 1
Output: "A"
Example 2:
Input: 28
Output: "AB"
Example 3:
Input: 701
Output: "ZY"
Solution (Incorrect)
At first glance, I thought it is just a simple conversion to a base 26 number. The obvious solution is:
string convert(int n) {
string s = "ABCDEFGHIJKLMNOPQRSTUVWXYZ";
string result = "";
int r;
while(n) {
r = (n  1) % 26;
n /= 26;
result = s[r] + ret;
}
return result;
}
However, it is incorrect. When n
is 27
, instead of AA
, it returns BA
. Hmm, that is because 271
is 26
,
which in 26 bassed numbering system carries one digit to the higher position and becomes 10
.
In our wrong notation system, where B
is equivalent to 1
, and A
is equivalent to 0
, 10
is noted as BA
.
Analysis
If we observe the question more closedly, we can find there is something special to the notation. It does not have
a notation for 0
! It is like in decimal, if we count 1, 2, 3, …, 9, then the next one, instead of 10, we
couldn’t write it down, because there is no 0
!
Hmm, this question must be created by an ancient mathmetician before the Indian invented 0
.
So how can we fix it?
Let’s study the current decimal system that has 0
. In decimal, a number 234 is actually
2 x 100 + 3 x 10 + 4 x 1
. A number 240 is 2 x 100 + 4 x 10 + 0 x 1
. A number 108 is 1 x 100 + 0 x 10 + 8 x 1
.
The position in the number corresponding to how many powers of 10, and the number is the coefficient.
So 942
the number 9
at the position (from left to right, start with 0) 2
, means you need to times number 9
with 10
to the powers of 2
(100
),
and the number 4
at the position 1
, means you need to times number 4
with 10
to the powers of 1
(10
),
and the number 2
at the position 0
, means you need to times number 2
with 10
to the powers of 0
(1
).
position 2 1 0
  
v v v
number 9 4 2
^ ^ ^
  
Power of 10 2 1 0
Value 9 x 100 + 4 x 10 + 2 x 1 = 942
If we omit 0
, then we cannot tell the difference between 108 and 18, or 10008! You see how important 0
is in our numbering system?
What to do if there is no 0
?
Let’s look at the number little more closely to figure out if there is a solution for this. A number 240
means
2 x 100 + 4 x 10 + 0 * 1
, if there is no 0
, we can do this: 240 = 23[10] = 2 x 100 + 3 x 10 + 10
. Here I use
[10]
as a notation means 10
occupies the last position. There, no zero is needed! Let’s use a single symbol to
replace the [10]
, say, A
, then 23[10]
can be written as 23A
. and 108
can be written as A8
.
From 1 to 110 becomes:
1, 2, 3, 4, 5, 6, 7, 8, 9, A,
11, 12, 13, 14, 15, 16, 17, 18, 19, 1A,
21, 22, 23, 24, 25, 26, 27, 28, 29, 2A
...
91, 92, 93, 94, 95, 96, 97, 98, 99, 9A
A1, A2, A3, A4, A5, A6, A7, A8, A9, AA
This, is exactly what we need! All we need to do is when a position should be 0
, we borrow a 10
from the higher position and use our A
to note a 10
.
Solution
Now back to the original question, the solution become obvious. When the remainder is 0
, instead of use 0
, we borrow
26
from the higher position, and write an Z
instead of 0
.

