Math Origami Protocols Security

"Where is Waldo?", or "Security by Origami"

The Problem

A friend of mine gave me a riddle this morning regarding “Where’s Waldo?”. The riddle is as follows:

You and a friend play “Where’s Waldo?”. You solve the puzzle before your friend, and you want to prove to your friend you solved the puzzle, without giving him any hints. How do you do this?

Obviously, this is of course very reminiscent of zero knowledge proofs. A good zero knowledge proof will allow your friend to be convinced before he found Waldo himself, and even if he never finds him at all. A somewhat less interesting solution is a solution that will allow your friend to be convinced if and when he finds Waldo himself and can then verify your proof.

There are obviously many “intuitive” solutions, and I will not write them here… I will write here the second solution I thought of, but I consider it to be the more interesting one. However, this solution doesn’t solve the original problem – it only allows your friend to verify your solution once he solved it himself.

The Solution

Take a sheet of paper of identical dimensions to the picture, and mark a spot on it in the position where Waldo would have been on that sheet of paper. Fold that sheet of paper to some kind of origami animal, and give it to your friend. Once he solves the puzzle, he can open the folding, and see for himself that the point was marked correctly.

This is obviously not a very good solution. It is just a glorified note – you write down your solution on a note, and ask your friend not to peek until he solves it himself. So I came up with an improvement:

Agree beforehand with your friend on some (large) origami folding (such as a beetle). He shouldn’t know the instructions to fold it. Take a sheet of paper, and mark Waldo’s position on it (with a dot). Hold the paper above the fold, and mark on the fold (with another dot) the projection of the original dot on the folding. Now unfold the origami – you have a crease pattern. Give the crease pattern to your friend. When he solves the puzzle, refold the origami, and prove to your friend that the projection of the dot on the fold coincides with the dot on the picture. As an added bonus – your friend just learned how to fold another origami beast!

Of course, this solution isn’t water-tight. It also relies on crease patterns being hard to solve. It is mostly security by obscurity – but this time, ‘security by obscurity’ becomes ‘security by origami’. I just found that fascinating – that origami may be considered a function that is ‘hard to reverse engineer’ (even if it is not so) – such as a hash function. Origami does behave a little bit like a hash…

Tell me your original solutions to the problem.

Algorithms Math

Reachability on arbitrary maps

The other day I had an idea. What if you took a map of some country, and from each coordinate computed the time it took to any other coordinate. The basic assumptions are that you can drive on road by car from parking lot to parking lot, and from parking lot to another point you can get only by foot. You can also add trains, and airplanes, each having less routes, but are actually faster. So for example, traveling from city A’s center to city B’s center is quite fast, but traveling from city A’s center to some point near the road from A to B will take some time.

This is a bit similar to computations done by game AI’s to determine how to get from point A to point B on a game board.

Now let us assume assume we want to draw the resulting graph on some kind of a 3D map, where the distance of each point from other points is determined by the time it takes to reach from that point to the others. Actually, it might not always be possible to plot this graph on a 3D map as a friend pointed to me, as 3 neighboring points already lock our point in space. But let us say that you can plot this 3D map. A’s center and B’s center will be near each other, a bit like a paper folding – which is incidentally the most common depiction of ‘science fiction wormholes’ explanations to laymen. It was pretty nice to try and visualize this kind of map. It is also interesting to think what is the least number of dimensions required to plot a given map.

Although I couldn’t come up with ideas about how this kind of visualization might be useful, it was still fun to to think of.


Python bug in __getslice__

While working with Python, a co-worker found a puzzling behavior… He couldn’t get correct slices on an object he created. After puzzling over the problem, we found out the problem – __getslice__ seems to change its arguments: in a 32 bit argument, when the MSB is set, the argument is changed to 0x7FFFFFFF. After looking it up, it turned out that the bug was also present in the latest Python 2.5. It also turned out that __getslice__ is deprecated. Here is the link to the bug info. This was the first time I got to submit a Python bug. Kind of a mixed feeling.

[Update: This bug was closed. You can still read about it in the original bug info.]


Fantasy books for programmers

I read these two books some years ago, before I was in high school. The concept really appealed to me. For fear of spoilers, I will not say much, suffice it to say reading the books gave me a different perspective on certain things, and some new ideas. The books themselves are not really complicated, don’t have the best characters around, but are a pretty solid read.  When I found them online some time ago, it took me just a few hours to go through one of them.

I recommend.

And I almost forgot… The links. Have fun.

Algorithms Programming Philosophy Python

Proofreading and what's wrong with the heapq module

Consider the following problem:

you have a large book you would like to proofread, with many chapters (100+) and a few men (4) at your disposal. How would you distribute the chapters among the men, considering that each proofreader must get whole chapters?

Well, the obvious solution is just divide the number of chapters by the number of men, and give each proofreader an appropriate number of (randomly picked?) chapters. But what if the chapters or of varying length? Well, you could just distribute them randomly, but that just doesn’t feel right now does it?

I was asked by a friend to help him write a script to solve this, and quite quickly I came up with the solution:

Sort the chapters according to length (number of words), in descending order. Keep the proofreaders in a sorted order. While there are still chapters left – get the next longest chapter, and give it to the proofreader with the least total length. Rearrange the proofreaders.

This algorithm is a greedy algorithm and is also regarded as the straightforward way to solve this problem.

Well, this seems simple enough – In the case where there are more then a few proofreaders – well, the obvious solution is to use a minimum heap. Using a minimum heap in python should be easy – just import the heapq module, use heapify and various other functions, just as you would use sort, and you are home free.

Not quite…

Turns out, that using the heapq module isn’t so easy as it seems, or at least not as quick and simple as I would like it to be. While you can sort list using your own compare function, and also providing a key-function, the heapq module sorts items using Python’s standard operators. This means, that to use heapq properly to sort an object according to some property, you have to write an object wrapper that has __lt__ and similar functions defined. All this coding could be saved, if heapq had a key argument that could be used, or any other method to control the comparisons.

And what about the proofreading you ask? Well, it worked. It divided the chapters quite evenly, although we did end up using sort() repeatedly, as the number of proofreaders was small and we did not want to overly complicate matters.

This again emphasizes a something I’ve come to discover – being able to write something quickly is usually more important then later being able to run it quickly. If you write something in an environment that allows for quick coding – later if speed is required you’ll be able to optimize it quickly, or even change the design or the basic algorithm.

Algorithms Math Programming Python

Interesting links – 3

How to Write a Spelling Corrector

This was a very interesting article, with some interesting python code, and a good mathematical explanation. I enjoyed reading it. I got to it through, which appeared on Daily-Python.

Robert Lang’s Origami Page

Fascinating page. A few months ago I was getting back to doing some Origami (it is a good relaxation method, also helps your wrists after typing too much :). I was trying to find ways to create origami creatures with more then two limbs, a head, and a tail, and I was fascinated by pictures of origami foldings of intricate creatures with many limbs, such as spiders and dinosaurs. I was also trying to formulate for myself some of the mathematical laws for origami. So I started to look into the subject and found out about crease patterns, and among many other sites, I got to Robert Lang’s site. The subjects he writes about are very diverse – and include using ‘origami knowledge’ to fold airbags, a program to create complicated crease patterns, an origami simulator (something I wanted to write myself :), and so on. Really, a fascinating read.