Permutations with repetition using Excel

The back story

Recently I have been trying to limit my use of VBA. It’s not that I was addicted and needed to go to VBA-rehab, on the contrary, I still love VBA till death do us part. But the thing is, every time someone opens an Excel sheet with VBA macros they are reminded of Excel’s vulnerability and the risks of macros.

So I set out to make workbooks that do the same thing, but without VBA. Not always is this possible, or efficient to do so. But when it’s possible, it also comes with great performance and great stability. No code needs to be changed, ever. Of course there are also downsides. It’s not as flexible as VBA, so you’re stuck in a rigid framework that solves one thing and one thing only. But it does it so well, oh my.

My latest endeavor was with permutations. I needed something that would generate all permutations of the tokens F,C,R,A (don’t ask) with repetition. As some know I am an avid speed cuber, that is solving the Rubik’s cube for speed. And the Rubik’s cube is a permutation puzzle. So I have dealt with permutations quite a lot. For those who haven’t paid attention in math class: you should know that permutations come in two flavors: with repetition (Pr) and without repetition (P). The number of permutations (Pr) in these four tokens F,C,R,A is 4^4, or 256. That number is exponential, so it grows so fast that at 5 tokens you are at 3,125 permutations and at 6 tokens at 46,656 permutations. At a set size of 10 tokens you are at 10^10 or 10,000,000,000 (ten billion) permutations. VBA would surely choke on that number of statements to follow. Excel can handle 1 million rows, although I wouldn’t put it to the test with that.

A clean slate

I started out writing a VBA program, that generates all permutations (Pr). The funny thing is if you go online and expect to find a bunch of worked out examples of algorithms, you don’t. Almost all examples you find are about permutations without repetition, which is like working with real objects, since you can’t duplicate real objects. The lottery is a good example of this. The program worked, but it was slow, and cumbersome. So let’s drop VBA and try it without.

CaptureFirst, I set out with some settings (sheet ‘settings’). We define the pattern, and calculate the size of the set (using LEN), and the number of possible permutations (using Length^Length). These values will be used extensively in the functions to generate permutations.

The maximum length of the pattern is set to 5, which totals 3,125 permutations, an amount which Excel can handle in the blink of an eye. You could extend the grid to a width of 7, which would come to 823,543 permutations. I’d be interested to know how fast Excel would generate the output, and how big the file would become. If you try it out please let me know.

Formula frenzy

CaptureGo to the output sheet and look at the grid on the right. There you see in the top row a couple of simple formulas. They are to set the repeat cycle of that column. In the first column you see a token repeated once, in the second column 4 times, in the third column 16 times. We’re multiplying by Length to set the repeat cycle. This is an easy way to generate permutations of any set. Think of how it works with regular counting. You start with 0,1,2,3,4,5,6,7,8,9, and then go one digit to the left, and you repeat, but now in cycles of 10, so that 10,11,12,13,14,15,16,17,18,19 has exactly 10 times a 1. We’re using the same principle.

Our basic generating formula is as follows:


If you don’t know MOD, this is a function for modulo, also called the remainder after performing division. The MID function gets a character from a specified position in the string. The column with N is simply to count and use the MOD function properly.

By using absolute referencing we are now able to copy the cell E4 to all other cells in the grid, while keeping a properly working formula. With a pattern of length 4 we can ignore the last column, which is only needed for a pattern of length 5.

In the settings sheet you will see a width and a height. By selecting the range starting at E4 with that width and height, we get exactly all permutations in the set.

Alternatively, on the left, there is a table with all tokens concatenated in one string, for ease of use. The formula for this is:


Using FCRA as a pattern we can now see all 256 permutations in column A!

Perfect permutations

CaptureWell, permutations without repetition are actually a subset of permutations with repetition (P < Pr). In a permutation without repetition you don’t have any duplicates. So for the tokens F,C,R,A a valid Pr would be FFFF, but it’s not a member of P. You can only get a member of P by swapping original tokens. So e.g. FRCA is a member of P. That’s why the number of items in P doesn’t grow as fast as in Pr. Four tokens gets to 4*3*2*1=24 permutations. This is called a factorial.

If a token set does not contain duplicates we can easily filter out the permutations we need. E.g. in the set A,A,B,B we still get duplicates in the list, and so we can’t filter. But in F,C,R,A it’s quite possible using Excel. The formula used is a but difficult though, and requires some thought:


Here, OFFSET gives us a dynamically defined range, which is handy, because we don’t know how long the pattern is beforehand. In this formula, Length is the size of the set (aka the length of the string). Both OFFSET and MATCH return multiple values, so it’s impossible to split the formula into more cells, but just for clarity, let’s view it in condensed form:

=SUM(IF(FREQUENCY({FFFA}, {FA})>0,1))=Length

What it does is it totals the frequencies of each character in the set, so in this case it returns 2, and then checks to see if it matches the length (4). If it matches we have a permutation. Note this only works for patterns that have no repeating tokens, like FCRA.

Using named formulas we can simplify the long formula to:


…where tokens equals OFFSET(E4,0,0,1,Length)

Now we have a formula to detect permutations. Unfortunately we still have duplicates, because our table always has 5 tokens and we might have a shorter pattern, like our example FCRA. So we use an IF to detect empty cells in column A and we can now use Excel’s filter (On the ribbon choose Data, then Filter) to get all permutations.


permutations with repetition (320KB)

Using named ranges and worksheet functions in Excel VBA

Bold Brackets

A couple of days ago I saw something in an article on StackOverflow, that blew my mind. I can’t find the article anymore, but I do remember what this one neat trick was, that will for ever change your VBA. It’s called a named range, and I found out I had always been doing it wrong. So have you, most likely.

Have you ever written something like this?

s = Application.WorksheetFunction.Sum(Range("A1:A10"))

You thought you were quite smart, using SUM to add some values together that would have taken a loop in VBA. You petted yourself on the back, took a beer, and applauded yourself for you being awesome. Well, you’re not awesome. This is lame. You suck. Ok, maybe not, but watch this:

s = [Sum(A1:A10)]

It’s incredible! This gives the exact same result.  You may wish to prepend with a sheet name, so it’s an exact reference. You can use any kind of name inside the square brackets. So, if A1:A10 is named ‘records’ in Excel you could rewrite this code to

s = [Sum(records)]

Note that with this notation we don’t use double quotes around the name of the range.

Vanishing Variables

CaptureA quite mighty use for this, is that we can now write code with a lot less variables, if we let Excel do the work for us.

  1. make a new sheet
  2. name it ‘variables’
  3. make three columns: name, value, description

Now you can fill the table you just made with all kinds of settings, constants or calculations that you want to use in your elaborate VBA program.

I am a lazy teacher, and I have a lot of students, so I work a lot with short macros that can help me get more spare time. Here’s an example:

    For Each subfolder In FSfolder.SubFolders
        If subfolder Like "Student *" Then
            [studentNr] = Right(subfolder, 6)
            If Not FileExists([sClass]) Then
                FS.createFolder [sClass]
            End If
            FS.MoveFolder subfolder, [newFolder]
        End If
    Next subfolder

In line 3 I fill the named range ‘studentNr’ with a value taken from a folder, which contains a student number. In the next line, I check if a folder for that students’ class has already been made, and if not, I make the folder. The variable [sClass] does not exist in my code. It only exists in my variable table in Excel. The cell contains a VLookup function to find in which class this student is currently enrolled. Similarly I have a [Teacher] variable, also with a VLookup function. NewFolder is simply a concatenation and formatting, which is also easily done in Excel.

As you can see, the values for [sClass], [Teacher] and [newFolder] are filled automatically by Excel, and I don’t have to process anything.

Programming like this in Excel is a new paradigm. You don’t churn out all your code top-to-bottom as you used to. You create sheets with lots of calculations, lookups etc, and then you make a tiny program that links all this together. Excel can do some crazy fast, complex stuff, and you should never have to program those anymore!

Crazy Caveats

Well wasn’t that incredible? You may not be used to programming like this, and I recommend this method only for experts. You should be in full control of the worksheets, or otherwise someone will mess with your program. Also, when you are part of a team, you should make sure this ‘magic’ is elaborately documented in the code (e.g. in a header of the function mention which Excel named ranges are used).

Good luck.

Learn more: FastExcel Blog


The selection contains multiple data values… Merging cells in Excel.

This is a super powerfeature I have always missed in Excel: the ability to join the content of cells without losing any data. Sorry, what’s that?? Yes, I know about the ‘merge’ option. It’s lame! Try merging two cells that both have content. Excel answers, delightfully happy, that you will only keep data in the first cell and lose all of the data of the other cells, and go deal with it.



So, I decided to ‘deal with it’ and I present here several macros for your pleasure and entertainment that will solve this once and for all.

Merging, Joining and Combining

Well, that’s all the same of course. Let’s start with a simple macro to join a bunch of cells and put the result in the first cell.

Sub Join()
   Dim out As String
   Dim c As Range

   out = ""
   For Each c In Selection.Cells
      out = out & c.Value
   Selection.Cells(1, 1).Value = out
End Sub

How does it work? First, we define the variable out which will contain all content. Then we loop over all cells in the current selection and we combine their values using the concatenation operator “&”. Finally we put the value of out in the first cell of the selection.

If you have never created a macro before, you can add them by pressing ALT+F11. After that, make sure to copy this one to the personal macro workbook.

In some occasions I needed to merge, but also keep the data separated by either commas or newlines. So I made a slightly modified version with arguments. Note for noobs: you can’t run these like normal macros, you’d have to go into the VB editor and start them from the immediate window. Or, you can make a short calling-macro similar to the JoinWithBreaks below.

' join the contents of a selection of cells
' and put all of these together in one single cell
Sub Join(Optional s As String = "", Optional wrap As Boolean = True)
    Dim out As String
    Dim c As Range
    Dim n As Integer

    out = ""
    For Each c In Selection.Cells
        n = n + 1
        out = out & c.Value
        If n < Selection.Cells.Count Then
            out = out & s
        End If
    Selection.Cells(1, 1).Value = out
    Selection.Cells(1, 1).WrapText = wrap
End Sub

Sub JoinWithBreaks()
    Join vbNewLine, True
End Sub


Another problem closely related to the merge problem is when you have a sheet looking like this:

excel fill problem

excel fill problem

And let’s say it extends a long long way to the bottom for your archive of ten years. To create a usable Excel table and e.g. to create a pivot table your table needs to have values in every cell, not just the first one. Humans can reason that cell A3 also belongs to January, but computers and thus Excel cannot. Clicking on the autofill handle in cell A2 fixes this, but doing so for 100 cells is still very tedious work. For that, we can use the following macro, which will automagically fill every empty cell with the value right above it.

Sub Filler()
    Dim c As Range

    For Each c In Selection.Cells
        If IsEmpty(c) And c.Row <> 1 Then
            c.FormulaR1C1 = c.Offset(-1, 0).FormulaR1C1
        End If
End Sub

How to fix the document map in MS Word 2007

The Document Map! That horrendous, horrible, horrific horror! It never does what I ask it to, it never shows what I tell it to show, it constantly messes itself up beyond recognition. It’s like a crazy guy beating himself up, I keep thinking of Fight Club. The first rule of Document Map is you don’t talk about Document Map. And yet, I am going to talk about it.

Today I suddenly found out the magic trick, the silver bullet that fixes the problem that had haunted me for so long.

As any professional MS Word user, I use the map extensively, even though its quirks pester me constantly. It’s just to powerful to let go.

What are its problems?

  1. Outline works, as long as you have very strict usage of headings. A lot of documents fail this, especially if they aren’t created by yours truly (ahem).
  2. Titles show no outline numbering, even though it’s definitely there. This is more an annoyance then an outright problem.
  3. The map is not updated. Grrrrr.

It appears Microsoft has put some fuzzy logic in the document map. Now most of the time I don’t really care about MS’s guess at what I want to do. If I press TAB then most of the time I actually want an indent. If I type ‘Teh’ I do mean ‘The’. And of course, I know how to disable that behavior.

However, the document map seems to magically create non-existing headings from my text, and not in the farthest corners of options and settings is there mentioning of the document map doing this. It’s like an easter egg, but not a very funny one.

And then I did what I often do when cleaning up documents. I pressed “CTRL+Q+SPACE”. If you don’t know this trick, it’s the ultimate cleanup action. It removes all and every formatting. I had the map open because I was about to do some things with headings when I noticed after pressing ‘CTRL+Q’ that the item disappeared from the map, as it should because it wasn’t a heading! Then, by not continuing with CTRL+SPACE I retained the original formatting (bold in this case).

Before you select all text and press CTRL+SPACE remember it wil turn lists into regular paragraphs, so make sure to just do this on the items that need to be removed from the map. If that still doesn’t work continue to press CTRL+SPACE. You will lose formatting, which you should recreate using styles. And if that fails make sure to press the Normal style button to revert to the default formatting.

The second problem appeared to actually be a bug. A heading right after a page break will not show its numbering. Well, that’s minor and you can always select ‘page break before’ in the style settings.

The third was related to both 1 and 2, perhaps it won’t happen again. I hope so, I really do.

Try it out and let me know of any other problems you encounter with the document map.

Update: someone created a macro for fixing the document map!

Fantasy Selling

photo of earth from NASA

The earth is FLAT! Seriously.

Here is a little mind control technique I just thought of: selling/planting an idea into someone’s head by just telling random stuff. You’d have to make sure to put enough symbolism in it, so it enters the brain of the other person through the power of their own associations. Make sure to bring in scientists, religion, government and media. Those are powerful friends to convince people.

The earth is flat

Goal: sell the idea that the earth is in fact flat.
Script: “Hey, you know the whole thing about the earth being a globe is in fact a hoax? Yeah, I didn’t believe it at first, but apparently it’s true. So when you see that image from space it’s always a circle but there is no shadow, so you know it can’t be round. Also, in the bible nobody says the world is round. In fact, in Daniel 4:10, “he saw a tree of great height at the centre of the earth…reaching with its top to the sky and visible to the earth’s farthest bound”. If everyone could see the tree, the earth must have been flat. Also, research has shown that people believe anything on television, and the images from space usually don’t show a moving earth: it is still, fixed as the bible says too. The reason you see that same picture of the earth from space again and again is because the media and the government need to sell this idea so bad. That’s why they need to repeat it, otherwise we would quickly fall back to the truth, which is that the earth is flat. “

A physics model of a physics model

One of the hardest topics in school next to computer science is physics. Time and again I see students struggle with the topics, and I have to admit I didn’t find it easy in high school myself. So I started studying physics again using Khan Academy, and gained much insight into the basics. I have admired Sal Khan since he started his site, and as a school teacher I also think I should help him, and my students, wherever possible.

I started drawing a diagram of how all things in physics connected to each other, and while doing that, I realized there should be software to do this better. After a short search I found the wonderful D3 toolkit, based on web standards, just as I like it. D3 can create awesome diagrams, graphs, infographics, you name it.

So now I have for your pleasure and learning aid a physics based overview of common physics quantities. Click the image to see the actual animated model.

Some tips on how to use it:

  1. Click the diagram to automatically redraw it
  2. Drag nodes to change the layout
  3. Mouseover a node to see related info
force diagram physics

force diagram physics

Use in physics assignments

The colors in the diagram denote related quantities:

  • blue – essential axiomatic units m, l, t
  • green – related to movement
  • orange – related to energy
  • red – related to a single unit

To use this diagram in your physics assignment, look at the known variables. Find those in the diagram. Now see if you can combine those to make more variables: follow the arrows. If you can start from two known values you can usually calculate a third. You continue doing this until you have found the desired outcome.

Salman Khan is the smartest man in the world

The world is changing. At some point in the future, says Ray Kurzweil, computers will be more intelligent than humans. He thinks it will happen in the year 2045 to be exact. That’s a scary thought, but he could be right. Many say he is the smartest man in the world.

But today I realized that Salman Khan is ahead of him. Khanacademy is an online tutorial for everything including math, biology, economics, chemistry, physics, history, and more. And Khan made all videos himself.

But not only did he do that, he is expanding at an exponential rate, to offer his courses to classrooms. And that could revolutionize education.

And all of that… for free, because he is filthy rich anyway, and doesn’t care about making more.

World changing: Khanacademy. See for yourself.

Facebook limits you at 5000 friends

enough is enough

enough is enough

Jeffrey Zeldman claims 5000 friends is not enough!

Not Amen! Agree to disagree!

The value in a social network lies in social. So the friends should be people you have met in real life and value in your social circle, which revolves around your life, not your business. A page is a business tool so that is what you should have.

The cap may be nonsense, but unfortunately there is no way for Facebook to validate that you actually know the ‘friend’ personally.

One other side-effect in which Facebook bites it’s own hand is that apps and games (OK and businesses) thrive by the amount of ‘friends’ you have. That is an incentive to make fake friends. I started to play CityVille only to find out that none of my friends did. If I were true to Facebook’s dogma I would quit playing or be an obnoxious friend who would evangelize CityVille to all my friends all the time, until they unfriend me. Now I play it happily with over 30 fake friends whom I’ve never met. I even block them from my wall, as I don’t want them in my real social circle. Once I stop playing CityVille I will quickly unfriend all these people.

Still, I fully agree that Facebook should work on a solution, which in my opinion is definitely not removing the cap but some way to have people in your network for a certain purpose, and have great distinction in how you treat and communicate with them.

All the positive talk about Twitter’s agility: don’t become so vigilent, or should they also remove the 140 character cap!?!?!?

I have always been appalled at the nomenclature of new applications: Friend and Like come to mind. I call them ‘close-related-marketing-profile’ and ‘increase-marketing-keyword-potential’.

Jeffrey, you don’t get to  5000 friends without making a few enemies :-P

Will Readability 2.0 change the web?

Jeff Zeldman writes a compelling argument for Readability 2.0. Readability is an open source tool that transforms a web page to a reading page, removing all clutter, including ads. Readability 2.0 will now actually pay the content providers (website owners) every time a user switches on Readability to read their pages.

At first I thought: Does this mean publishers don’t need to sign up?? So they get an unexpected check from a seemingly philanthropic web company?? But after looking at the site, I realize publishers do need to sign up. And that, I think, makes it quite unpractical. Let me tell you why.

How about promoting Readability on a site with ads? That’s like saying: click here to screw with all our other clients. So that’s virtually impossible.

I believe we can divide the site owners in four groups

1. The small site owner, not being able to get advertisers easily, not making enough on Google Ads. They could promote Readability, but wait: there are no annoying ads on the page. Duh. What’s the point? Well, to be honest, they could litter the site with Google Ads and then have a shiny button that says : “don’t like ads? click here to remove them”. The user would then learn about readability, sign up and start paying.

2. The large site owner who is in a clinch with advertisers, and says: we’ll just move our whole platform to readability, @#$% you very much. You pay Readability, after which you get a ‘pro’ account which gives you access to more content. That would require a partnership with Readability I guess, unless there is a technical solution (checking to see if you are using Readability).

3. The large site owner who has littered the site with ads, who likes Readability too, but is unable to promote it, fearing the wrath of advertisers. There’s no way for this owner to put a button on the site.

4. Like, the large site owner who is not in it for the money, or actually, makes enough doing other stuff and just uses the site for self-promotion. They don’t have ads, and using Readability is nice, but unnecessary.

I am dying to see some stats soon, so we can see if offering Readability instead of advertisement is a valid business decision. If it is, it will spread and transform the web. At some point Readability then needs to be taken over by a big player, or it will be simply sued to death by angry advertisers.

Why is writing for the web difficult?

Blogging for dummies

I have been a web professional for over 10 years. In that time I have learned a lot of things: PHP, CSS, XHTML, JavaScript, XML, Photoshop. But one of the things that has always been harder to do right is writing for the web. The reason? It has so many conflicting motives, that it’s impossible to have one good writing style. Let’s analyze:

When we write for the web we most certainly write for the fast paced audience that has no time to read pages and pages of text. That’s a given, and it’s what sets it apart from writing for paper. The other characteristic is links: if you need to explain something, it’s easier to just link to wikipedia than to insert the actual text. Also you don’t want to infringe on copyright.

Let’s look at our goals for writing for the web:

  1. Make it easy to read
  2. Make it easy to use
  3. Make it easy to find (search engine position or ranking)
  4. Make it sell (in case you’re selling, but you’re always selling ideas)
  5. Make it look good

Those goals are also connected to viewpoints from certain people: the writer, the usability expert, the SEO specialist, the sales representative and the designer. These fantastic four have to make sure the page adheres to all their standards. And here is where it starts to itch:

  1. easy to read: plain fonts, short sentences, short text, clear message, bold, pictures and diagrams that explain difficult topics.
  2. easy to use: minimal use of text, clear and large buttons, minimal use of design elements
  3. easy to find: clear headings, short sentences, keywords in bold (not the same ones), no pictures required.
  4. sell:  everything leads to a buy, no navigation on cart page
  5. look good:  creative fonts, no bold text, plenty of  non-illustrative pictures (preferably photographs) and design elements

In my opinion you should work from the outside in: make it look good, then make it easy to use, easy to read, easy to find and easy to sell, in that order. But as you make it look good (design process) you can of course have some consideration for the other aspects. It should also be noted that SEO is still voodoo, since nobody knows how Google really works. Also, search engines, in particular Google, change their ways of working constantly and they get better and better at identifying your pages. So don’t try to fool them, it’s not worth it.

There will probably always be a battle between usability experts and designers, because their worlds are so far apart. But to a modern web user something like looks like it was made 20 years ago and doesn’t instill trust in a user, something that is vital to sales.

Remember I may not be an expert in all those areas,  but the ideas presented do come from the leading experts in these areas.


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