How to sort table using function/equation only

Okay, here's the "one formula does it all" solution without additional temporary columns:

Formula in A6:

=INDEX($A$2:$E$2,MATCH(SMALL($A$3:$E$3+COLUMN($A$3:$E$3)/100000000,COLUMN()),$A$3:$E$3+COLUMN($A$3:$E$3)/100000000,0))

Enter it as an array formula, i.e. press Ctrl-Shift-Enter. Then copy it to the adjacent columns.

To get also the number, use this formula in A7 (again as array formula):

=ROUND(SMALL($A$3:$E$3+COLUMN($A$3:$E$3)/100000000,COLUMN()),6)

Both formulas are a bit bloated as they have to also handle the potential duplicates. The solution is to simply add a very tiny fraction of the column before applying the sorting (SMALL function) - and then to remove it again...

I think an easier solution is to use the "Large" function:

The Column functions are just an easy way to count from 5 down to 1, but a helper column may be even easier.

You can have a similar answer using the "Small" function as well.

Although I have used the "add a small number technique", I think this is the most elegant for a helper row/column:

=RANK(B4,$B$4:$F$4,0) + COUNTIF($B$4:$F$4,B4)-1

(copy across the row column) RANK gets you close and the COUNTIF portion handles duplicates by counting the number of duplicates of the cell to that point. Since there is always match (itself) you subtract 1 for the final rank. This "sorts" ties in the order that they appear. Note that an empty cell will generate #N/A and character data as #Values.

It's doable! :-)

Here is the sample file.

Explanation

I assume that your data is in row 1 of Sheet1 and that you want the data sorted starting in column 1 of another sheet - if not, adjust accordingly.

Place this formula in column 1 of the target sheet, say in row 2:

=ROUND(SMALL(Sheet1!$A$1:$E$1+COLUMN(Sheet1!$A$1:$E$1)/100000000,COLUMN()),6)

You need to enter the formula as an array formula, i.e. press Ctrl-Shift-Enter.

In case you place it in another column than column one, you need to replace COLUMN() with COLUMN()-COLUMN($A$2)+1 where $A$2must be a reference to the cell itself.

This formula will return you the lowest number in your range - if you copy it the next 4 columns, you'll get your order list of numbers.

To translate this back to the column number, we need to perform 2 steps:

1. Figure out the column number:
   This can be done with with this formula:

   =MATCH(SMALL(Sheet1!$A$1:$E$1+COLUMN(Sheet1!$A$1:$E$1)/100000000,COLUMN()),Sheet1!$A$1:$E$1+COLUMN(Sheet1!$A$1:$E$1)/100000000,0)
   

   - again, to be entered as array formula. If the source data does not start in column A, you need to add +COLUMN(Sheet1!$A$1)-1 where $A$1 needs to be replaced with the leftmost cell of the source data.
2. Convert the column number to a letter:
   Assuming your column number from step 1 is in cell A6, this formula will do the job:

=LEFT(ADDRESS(1,A6,2),SEARCH("$",ADDRESS(1,A6,2))-1)

Of course, you can also combine step 1&2, this will result in this megaformula:

=LEFT(
    ADDRESS(
        1,
        MATCH(
            SMALL(Sheet1!$A$1:$E$1+COLUMN(Sheet1!$A$1:$E$1)/100000000,COLUMN()),
            Sheet1!$A$1:$E$1+COLUMN(Sheet1!$A$1:$E$1)/100000000,
            0),
        2),
    SEARCH(
        "$",
         ADDRESS(
             1,
             MATCH(
                 SMALL(Sheet1!$A$1:$E$1+COLUMN(Sheet1!$A$1:$E$1)/100000000,COLUMN()),
                 Sheet1!$A$1:$E$1+COLUMN(Sheet1!$A$1:$E$1)/100000000,0),
             2)
     )-1
)