Spreadsheet Spelling Test

The A-to-Z system is a base-26 number system. Let $A=1$ , $B=2$ , $C=3$ $\cdots$ $Y=25$ and $Z=A0=26$ . Therefore,

$\begin{aligned} EXAM_{26} & = (E \times 26^3 + X \times 26^2 + A \times 26 + M)_{10} \\ & = 5 \times 26^3 + 24 \times 26^2 + 1 \times 26 + 13 \\ & = 87880 + 16224 + 26 + 13 \\ & = \boxed{104143} \end{aligned}$

If we have A - Z (the rightmost letters, or least significant letters) the numbers will be 1 - 26. So 26 columns in total.

If we then have AA - AZ, that will take 26 columns, so each letter in the second place value will each have 26 columns to go through.

So to get from AA - ZZ

We will have to get through $26 \times 26$ or $26^{2}$ columns

We can then express this is a place value problem. We have to assign numbers to the letters. We have to shift the numbers up by 1 so A = 1, and Z = 26. This is because if A is 0, then AA would yield 0.

If we spell EXAM with the spreadsheet...

$E=5, X=24, A=1, M=13$

$5\times26^{3}+24\times26^{2}+1\times26+13=\fbox{104143}$

You can try this on a spreadsheet program like Excel! Try a word like CAT and see if you get the right number!

Leave comments if you find a more efficient way of doing this.

Well, I just do it step by step:

From A to Z $\Rightarrow$ $26$ numbers

From AA to ZZ $\Rightarrow$ $26\times 26$ numbers

From AAA to DZZZ $\Rightarrow$ $5\times 26\times 26\times 26$ numbers

From EAAA to EWZZ $\Rightarrow$ $23\times 26\times 26$ numbers

From EXAA to EXAM $\Rightarrow$ $12$ numbers

Totally, there are $\begin{aligned}26+26\times 26+5\times 26\times 26\times 26+23\times 26\times 26+12=\boxed{140143}\end{aligned}$ numbers.