few problems, approaches and solutions

## Four Cards, Gnome, and Goblin

Another interesting puzzle, Four Cards, from here.

Four cards are placed on a square table, one card at each corner. A blind gnome and an evil goblin take turns to play the following game. The blind gnome gets to choose a subset of the four cards and flip them simultaneously. If all four cards are face up, he wins the game. Otherwise, the evil goblin get to rotate the table by an amount of his choice. Show that for any initial configuration of cards chosen by the evil goblin, the blind gnome can win the game in a finite number of steps with a deterministic strategy.

To get a better start with the problem, let me simplify it a little. In the modified problem, the gnome wins if all cards face the same. Example, either all cards face up, or all cards face down. This is a simpler problem.

Now lets take the simplified version. Consider the 4 cards on the table. Removing all the symmetrical configurations, there are just three distinct configurations of cards on the table, apart from the winning configuration. For notational convenience, I will use D for a cards facing down, U for a card facing up.

```Case 1: Just one face differs from others.
|D D|
|D C|

Case 2: two cards up and two down on the same side.
|D C|
|D C|

Case 3: Two cards up and two down diagonally.
|D C|
|C D|
```

The gnome has to know about which kinds of flips matter. He cant judge about which cards to flip, so he only has the option of differentiating flips by count and orientation. Again, there are just 3 kinds of flips that matter. Flipping one of them, flipping 2 diagonally, or flipping 2 sideways.

Now map the transformations possible by these flips on the 3 types of configuration.

```1 -diag-> 1
1 -side-> 1
1 -one->  {2, 3, win}

2 -diag-> 2
2 -side-> {3, win}
2 -one->  1

3 -diag-> win
3 -side-> 2
3 -one->  1
```

Lets map the transformations for a the winning game. We would have to keep track of set possible cases that will occur in each transformation. Observe that case 1 is most distant from wining, case 2 the second most, and case 3 the closest. What we would like to achieve is a way to reduce the possible number of configuration, as we work with transformations. So if there is a case 3, finish off with it and so on. If a win happens in between, game ends automatically. So we only have to consider in the cases otherwise. Here’s the solution

```{1,2,3} -diag-> {1,2} -side-> {1,3} -diag-> {1} -one-> {2,3} -diag-> {2} -side-> {3} -diag-> {sure win}
```

To the main problem. If the only winning position is all faces up, the game essentially remains the same except the number of configurations and transformations. Now the number of combinations that matter is 5. The types of transformations remain the same, except for the special case of flipping all the cards. Its only used in one case. For rest of the configurations, flipping all doesn’t change anything. Here’s a list cases and the transformations which are useful. I am showing win case if its the only case.

```Case 1:
|U U|   1 -diag-> {1,2} ; 1 -side-> {1,2} ; 1 -one-> {3,4,5}
|U D|

Case 2:
|D D|   2 -diag-> {1,2} ; 2 -side-> {1,2} ; 1 -one-> {3,4,5}
|D U|

Case 3:
|U U|   3 -diag-> {3} ; 3 -side-> {4,5} ; 3 -one-> {1,2}
|D D|

Case 4:
|U D|   4 -diag-> {5} ; 4 -side-> {3} ; 4 -one-> {1,2}
|D U|

Case 5:
|D D|   5 -diag-> {4} ; 5 -side-> {3} ; 5 -one-> {2} ; 5 -flipall-> {sure win}
|D D|
```

Now the solution for the original problem. What we would try to do it get all the cases to case 5, and then flip-all. Flip all is represented by fa.

```{1,2,3,4,5}
-fa->   {1,2,3,4}
-diag-> {1,2,3,5}
-fa->   {1,2,3}
-side-> {1,2,4,5}
-fa->   {1,2,4}
-diag-> {1,2,5}
-fa->   {1,2}
-one->  {3,4,5}
-fa->   {3,4}
-diag-> {3,5}
-fa->   {3}
-side-> {4,5}
-fa->   {4}
-diag-> {5}
-fa->   {sure win}
```

Essentially, what we did is to take the following action at each step.
if 5 one of possible states then flip-all
else if 4 among the possible states flip diagonal
else if 3, flip side
else if 2 or 1, flip one

So gnome can win the game in 15 steps or less.

## splitting pdf slides: 4-up to 1-up

Many 4-up slides are quite inconvenient for onscreen reading. And quite a few times, no other options are available. Here is a small and rough solution to the problem. First, you need to have Python on your system. Download and extract the pyPdf package. Download code from here, to the source directory of the pyPdf package and run it from there. To use it give input and output file names and number of slides per page (<row> <col>) on the input pdf, and the program should work. If the slides don’t show up in proper order, give the order of slides you want, at the end of command line.

If you would like to install the pyPdf package, you would have to copy page4eachXobj function to PageObject in pdf.py and add ‘import copy’ to the top of the file. While copying, be careful with indentation. After this, my file could be placed anywhere.

The file has two methods to convert 4-up/6-up/… slides to a more convenient one slide per page. First method adv_slide_split gives better output(ex: similar to original slides), but works in very few cases. This method doesn’t use the row column information, given at input. Second method slide_split works by dividing the page into equal sized pages, usually not as good as the original 1-up, but works with most of the cases. If possible, the program will use the first method, else the second one.

Rest of the post is about my experience with the problem. Not a required read for using the program.

First off, you need to know that pdf is, in simple words, a collection of data and instructions on how to present the data. Its different/complex from markup languages, as it has to deal with various devices, like printer, and not just the web. And has to be able to represent complex typography and graphics etc, in a concise way.

A pdf file contains lots of <num> 0 obj and endobj pairs in it. Called Indirect Objects, these are used as a reference to objects in the file. Example, from collection of pages to postscript instructions and fonts and colors etc. There are few kinds and types of objects. Name objects starting with ‘/’, dictionary objects enclosed in ‘<<‘ ‘>>’ pair, array objects in ‘[‘ ‘]’ pair, stream object mostly used for data. For our purpose, these should be enough. Indirect objects making it easy to share objects among objects! <num> 0 R is used to refer to an object. I have skimmed, skipped and missed much of the information. If you are interested, read the manual [4.5Mb]. Chapter 3 is the most relevant part.

I was lucky enough to stumble upon a nicely generated slide to dig into, similar to this one . A casual look into the raw file shows /Page tags.

```7 0 obj <<
/Type /Page
/Contents 8 0 R
/Resources 6 0 R
/MediaBox [0 0 841.89 595.276]
/Parent 9 0 R
>> endobj
```

With the /Page tags are /Resources with an object reference. The referred object is /XObject dictionary with references to 4 objects.

```6 0 obj <<
/XObject <</Im2 2 0 R /Im3 3 0 R /Im4 4 0 R /Im5 5 0 R>>
/ProcSet [ /PDF ]
>> endobj
```

I removed 3 of them from the dictionary. The pdf file indeed showed only the remaining slide (with empty space for others), while xpdf complained about missing references. Manual mentions that /XObject names external objects. That means the names must be used somewhere else. But searching for those names show only the /XObject entries. Page objects entry in manual mentions that /Contents is where the contents of page are described. But /Contents refers to a /FlateDecode stream (essentially a zipped binary of the description of page). Fortunately, I didn’t have to look inside it. Need was just to display those external references, one on each page. The external references in /XObject themselves were of type /XObject.

```2 0 obj <<
/Type /XObject
...
>>
stream
...
endstream
endobj
```

Manual mentions details about the type and how to display it. To have a proof of concept, I changed a Page objects /Contents key to point to a custom stream, and /Page to contain one slide. Here’s the output. More work was needed to convert the whole document. New pages would be needed, as each page would have to be replaced with 4 new once, references would have to be updated, page counts at all levels need to be updated, xref would have to be updated etc.

This needed considerably more effort than I intended. Again, fortunately, I came across pyPdf. Its a versatile library. In the README itself, it shows the trick of cropping a page (Although I would call it zooming). So all I had to do was to copy page 4 times and only show relevant part in each of them.

The above method doesn’t depend on slides being available as separate resources, but used the concept of MediaBox. So it would work with most pdf files. But, if the slides are available as separate resources, the earlier method results in a pdf closer to the original sources. For comparison of the output, here are slides generated with method using XObject and the MediaBox one. Fortunately the library also makes it easier for me to implement the earlier /XObject method. It depends on the internals of pyPdf implementation. So either you would have to copy it to the pyPdf source file, or import all the internal classes of pyPdf.

In all, this is less than a days (cumulative) work.