Formula Forensics 006. Palindromes

Formula Forensics 006. Palindromes:

Chandoo wrote a post in August 2011 where he looked at determining if a cell contained a palindrome.
Chandoo presented a formula for determining if a cell; C1; contains a palindrome:
=IF(SUMPRODUCT((MID(C1,ROW(OFFSET($A$1,,,LEN(C1))),1)=MID(C1,LEN(C1)-ROW( OFFSET($A$1,,, LEN(C1) )) +1, 1))+0)=LEN(C1),"It’s a Palindrome","Nah!")
And then Chandoo challenged everyone:
How does this formula work?
Well, that is your weekend homework.
So today we’re going to complete our homework and pull apart the above formula and see what makes it tick.

Palindrome Setup

Download the example file so you can follow along with a worked example, Excel 97-2010.
In a blank worksheet enter
C1: “Chandoo” without the brackets
D1: =IF(SUMPRODUCT((MID(C1,ROW(OFFSET($A$1,,,LEN(C1))),1)=MID(C1,LEN(C1)-ROW( OFFSET($A$1 ,,, LEN(C1)))+1,1))+0)=LEN(C1),"It’s a Palindrome","Nah!")
In the example below we will work on two words:
Firstly, “Chandoo” which is clearly not a Palindrome
and Secondly, “Radar” which is a Palindrome

Non-Palindrome Analysis

The main structure of this formula is that it is a simple If () function.
The Excel If() function is defined by 3 parts
=If(Condition, Value if True, Value if False)
Our Formula is
=IF( SUMPRODUCT(( MID(C1, ROW( OFFSET( $A$1,,,LEN(C1))), 1) = MID(C1, LEN(C1) - ROW( OFFSET( $A$1,,,LEN(C1))) + 1, 1)) + 0) = LEN(C1), "It’s a Palindrome", "Nah!")
Condition:                 SUMPRODUCT((MID(C1,ROW(OFFSET($A$1,,,LEN(C1))),1) = MID(C1, LEN(C1) - ROW(OFFSET($A$1,,,LEN(C1)))+1,1))+0)=LEN(C1)
Value if True: "It’s a Palindrome"
Value if False: "Nah!”
Lets not waste time on the two Values if True/False as they are purely a message to the user, the real work happens in the Condition part of the If() function.
The Condition does all the work: SUMPRODUCT((MID(C1,ROW( OFFSET($A$1,,, LEN(C1))),1) = MID(C1, LEN(C1)-ROW( OFFSET($A$1,,,LEN(C1)))+1,1))+0)=LEN(C1)
Is a Sumproduct and a Len
That is the formula is doing a calculation of the sumproduct of some inner calculations and comparing the answer to the length of the contents of cell: C1 in the example of "Chandoo" = Len(C1) = 7



SUMPRODUCT(( MID(C1,ROW(OFFSET($A$1,,,LEN(C1))),1)= MID(C1, LEN(C1)-ROW( OFFSET($A$1,,, LEN(C1)))+1,1))+0)=LEN(C1)
Lets now look at the two inner calculations:
MID(C1,ROW(OFFSET($A$1,,,LEN(C1))),1)
Copy this calculation into E7
= MID(C1,ROW(OFFSET($A$1,,,LEN(C1))),1) Don’t press Enter, press F9
(If using the example file goto cell E9, Press F2 and then F9)


Excel will return {"C";"h";"a";"n";"d";"o";"o"} Don’t press Enter, press Esc when ready
It is an Array of the letters in the cell C1
Now do the same for the second part of the equation:
Copy this calculation into E8
= MID(C1, LEN(C1)-ROW(OFFSET($A$1,,,LEN(C1)))+1,1) Don’t press Enter, press F9
Excel will return {“o”;”o”;”d”;”n”;”a”;”h”;”C”}
You will notice that this array is the reverse of the word in C1
We will come back to how these two equations work in a minute or two
But note that we now have
Sumproduct(({"C";"h";"a";"n";"d";"o";"o"}={“o”;”o”;”d”;”n”;”a”;”h”;”C”})+0)
We can evaluate the inner part of this to see what happens
Copy this calculation into E10
={"C";"h";"a";"n";"d";"o";"o"}={“o”;”o”;”d”;”n”;”a”;”h”;”C”} Don’t press Enter, press F9
Excel will return an Array {FALSE;FALSE;FALSE;TRUE;FALSE;FALSE;FALSE}
This means that only the 4^th letter or “n” in Chandoo is the same forward and backwards.
Looking at the next bit
Copy this calculation into E12
=({"C";"h";"a";"n";"d";"o";"o"}={“o”;”o”;”d”;”n”;”a”;”h”;”C”})+0 Don’t press Enter, press F9
Excel will return {0;0;0;1;0;0;0}
Excel has converted the false/True array above into an array of 0′s and 1′s


Finally in E14 evaluate:
=Sumproduct({0;0;0;1;0;0;0})
Is evaluated and Excel adds up all the numbers returning a 1 as the answer.
So the original equation :
=IF(SUMPRODUCT((MID(C1,ROW(OFFSET($A$1,,,LEN(C1))),1)=MID(C1,LEN(C1)- ROW( OFFSET($A$1,,, LEN(C1))) +1,1)) +0 )=LEN(C1),"It’s a Palindrome","Nah!")
Is simplified as
=IF( 1 = LEN(C1),"It’s a Palindrome","Nah!")
Now C1 has the Word “Chandoo “ in it which is 7 letters long
=IF( 1 = 7,"It’s a Palindrome","Nah!")
So the If() function returns the False answer of “Nah!”

Non-Palindrome Analysis

If we place a Palindrome such as “Radar” in C1 and skip backwards to the
SUMPRODUCT(( MID(C1,ROW(OFFSET($A$1,,,LEN(C1))),1)= MID(C1, LEN(C1)-ROW(OFFSET($A$1,,,LEN(C1)))+1,1))+0)=LEN(C1)
Section , Evaluating each part again we see
E21: MID(C1,ROW(OFFSET($A$1,,,LEN(C1))),1)
Evaluates to {"R";"a";"d";"a";"r"}
And
MID(C1, LEN(C1)-ROW(OFFSET($A$1,,,LEN(C1)))+1,1)
Evaluates to: {"r";"a";"d";"a";"R"}
And
( MID(C1,ROW(OFFSET($A$1,,,LEN(C1))),1)= MID(C1, LEN(C1)-ROW(OFFSET($A$1,,,LEN(C1)))+1,1))
Evaluates to: {TRUE;TRUE;TRUE;TRUE;TRUE}
With
SUMPRODUCT(( MID(C1,ROW(OFFSET($A$1,,,LEN(C1))),1)= MID(C1, LEN(C1)-ROW(OFFSET($A$1,,,LEN(C1)))+1,1))+0)
Evaluating to: 5
Which is clearly equal to the length of the word “Radar” and so the If() function returns “It’s a Palindrome”

A Minute Later

But how do the middle bits
MID(C1,ROW(OFFSET($A$1,,,LEN(C1))),1)
and
MID(C1, LEN(C1)-ROW(OFFSET($A$1,,,LEN(C1)))+1,1)
Work?
The two equations are effectively the same
The first works left to right and extracts each letter one at a time
The second works right to left and extracts each letter one at a time
How Does
=MID(C1,ROW(OFFSET($A$1,,,LEN(C1))),1) F9
Evaluate to {"C";"h";"a";"n";"d";"o";"o"}
=MID(C1,ROW(OFFSET($A$1,,,LEN(C1))),1) F9
Is a simple Mid() function which takes the 1 Character at position ROW(OFFSET($A$1,,,LEN(C1))) from the contents of C1
What is ROW(OFFSET($A$1,,,LEN(C1)))
Is used to return an array of numbers from 1 to Len(C1) in this case 7
eg: {1;2;3;4;5;6;7}


So in E16 enter =ROW(OFFSET($A$1,,,LEN(C1))) and press F9
Excel evaluates it to {1;2;3;4;5;6;7}
So the function =MID(C1,ROW(OFFSET($A$1,,,LEN(C1))),1)
Returns the 1st, 2nd. 3rd, 4th, 5th, 6th & 7th characters from C1 as an Array

How Does =ROW(OFFSET($A$1,,,LEN(C1))) work?

=ROW(OFFSET($A$1,,,LEN(C1)))
Takes the Row of the range defined by the Offset Function
Note that OFFSET($A$1,,,LEN(C1)) is a simple Offset that sets up a range
The excel Offset Function is defined as
=Offset(Reference, Rows, Columns, [Height], [Width])
In our example
=OFFSET($A$1,,,LEN(C1))
Will return a Range which is referenced to A1, has no Row or Column offset and is the length of cell the contents of Cell C1
Effectively returning a range A1:A7
Because this is all in an Sumproduct formula, Excel evaluates this formula for each value in the Range
And so
=ROW(OFFSET($A$1,,,LEN(C1)))
evaluates it to
{1;2;3;4;5;6;7}
Which is then used extract the characters from the word in C1 into an Array as {"C";"h";"a";"n";"d";"o";"o"}

A similar process applies to the second half of the two equations
MID(C1, LEN(C1)-ROW(OFFSET($A$1,,,LEN(C1)))+1,1)
Except that it is evaluated from the Right to Left of the Word in C1 by use of the
LEN(C1)-ROW(OFFSET($A$1,,,LEN(C1)))+1
In the Mid Equation

Summary

So in summary we use Sumproduct to compare two Arrays, which contain the word and the word reversed, to each other. The Sumproduct counts the number of common matches and then this is compared to the length of the word.
If the two match the word is a Palindrome.

DOWNLOAD

You can download a copy of the above file and follow along, Download Here.

OTHER POSTS IN THIS SERIES

You can learn more about how to pull Excel Formulas apart in the following posts which are all included in the Formula Forensic Series:

WE NEED YOUR HELP

I am running out of ideas for Formula Forensics and so I need your help.
If you have a neat formula that you would like to share and explain, try putting pen to paper and draft up a Post as Luke did in Formula Forensics 003 or like above.
If you have a formula that you would like explained but don’t want to write a post also send it in to Chandoo or Hui.