This article explains a general method that can theoretically be used to solve any burr puzzle, starting from the set of disassembled pieces, and knowing only the final shape.
After a brief overview of the method, we will apply it step by step to the solving of the Puzzle #23, designed by Derwin Brown. It is a 6-piece cross that needs 6 moves for the first piece to be removed.
In practice, the method works very well with six-piece crosses. It takes between 5 minutes and one hour to solve one this way. For burrs with more pieces, the method can be very long and quickly becomes unusable.
Actually, unless a burr puzzle has been specially designed with symmetry, aesthetics or simplicity considerations, when the number of pieces increases, the time needed to solve it grows faster than exponentially. For a 36-pieces cross of mine, Burrtools, trying to find the solution, estimates the remaining computing time to "ages", which means more than one million years, the highest time that it can display.
Overview
The goal is to try every possibility, without missing any, and without trying twice the same. This way, if the puzzle has a solution, we will necessarily find it.
For this purpose, we are going to try all the pieces, from the first to the last, in all the positions, from the first to the last, in all orientations, from the first to the last.
When the pieces can fit in the final shape in a given position, it does not necessarily mean that we have found the solution. Some of such assemblies, as we call them, can be impossible to build in practice, because when all of the pieces are in place but one, the last one can't be inserted in any way.
Some puzzles can have many of these assemblies that are not solutions. We will have to go on trying the next possibilities until we find an assembly that can actually be build. Only then will the puzzle be solved.
Preparation
Here is a picture of Puzzle #23, that we are going to solve.
Since we are going to enumerate pieces, positions and orientations, we need a sorting order on which we can rely. Choosing a good sorting order is very important, because if we have the slightest doubt about the rank of a piece or an orientation in our list, we may skip some possibilities, and miss the solution.
Here is the sorting order that we are going to use for the possible positions inside the final shape.
These picture do not represent the real pieces, but only a skeleton. The idea is that position A will be a piece laying flat on the table, aligned with ourselves. Position B will be a horizontal piece perpendicular to A, on the far side. C will be the vertical piece on the left side. D the middle horizonal piece on the front side, E the vertical piece on the right side, and F the top horizontal piece.
This sorting order is completely arbitrary. It is chosen here just because the initial author of this page is used to it.
It is very important to choose an order that you can remember without hesitation. When you have 3 piece in hand, you must be able to tell exactly what is the last one that you tried, and where you are going to try the next one.
Sorting the positions is just a matter of practice. Sorting the pieces can be a bit more tricky. Here are the six pieces of Puzzle #23:
Being able to sort the pieces without error is extremely important. When you realize that the piece that you have in hand can't fit, you must be able to tell without hesitation which of the pieces left on the table were already tried, and which ones were not.
It is up to the puzzlist to choose a sorting order that he or she can remember easily. In Puzzle #23, the pieces are not easy to sort. The order chosen here relies first on the number of upper cubes near the extremities of the pieces (repsectively 3, 2, 1, 0, 0 and 0 for pieces 1 to 6), then on the presence of two levels of cubes in the middle of the pieces (respectively yes, yes and no for pieces 4 to 6), and on the number of holes in the back of the pieces (respectively 1 and 2 for pieces 4 and 5).
These criterions may seem a bit awkward, but it doesn't matter. Let's just assume that the pieces are numbered from 1 to 6.
And last, when we try a piece, we must try it in all possible orientations. Here, we will choose to try to position the backbone of the piece in the following order: on the left, then on the right. In the back, then in the front, or down, then up.If the back of the piece is full (like for piece 1), the criterion will be the side which has more cubes.
Execution
The method consists in taking the first piece, place in in the first position, in the first orientation. Then the second piece in the second position in the first orientation. If it is not possible, we try it in another orientation. If it doesn't fit, we put it down and try the third piece, then the fourth, etc. Each time in all possible orientations. Everytime a piece doesn't fit, we begin to look for another orientation. If it doesn't work, we put it back and try the next.
If we reach the last piece and it doesn't fit, then it means that the last piece that we tried was wrong, and we remove the pieces from the puzzle until we find a new possibility to try.
Let's apply this method to Puzzle #23.
Take the first piece, and place it in the first position.
We note this configuration A1, which means, position A, piece 1. We chose to put the side wich has most cubes on the back of the puzzle, but for the first piece, it doesn't matter since if we had placed it in front, the result would have been exactly the same, the whole puzzle being just rotated by 180°.
And as soon as we use one piece, we have a problem. This piece has an ambiguous orientation. We don't know which side is its back! The same happens with piece 2, shown here as an example:
As long as the side of a piece ends with two full squares, it is a valid side to use on the outside of the puzzle. The other pieces have not this problem, as shown with piece 3. We will thus have to try these pieces in four different orientations instead of two. Let's choose to sort them with the side that has the most cubes on the outside of the puzzle first. For piece 2, it would give
Which would mean Position A, Piece 2, Orientation a (A2a), then Position A, Piece 2, Orientation b (A2b).
But let's get back to piece 1. We have placed its full side on the table. This is configuration A1a. Now, let's take piece 2, and put it in position B. After having tried all possible orientations, it only fits in the last one: hollow side back, and backbone up, which is the fourth one, that we call orientation d.
Let's look more in detail how we proceed, with a simpler piece. We now take piece 3 and try to put it in position C, orientation a, that is with the backbone on the back side:
It doesn't fit. So let's try the other way, with the back bone on the front side:
It doesn't fit at all. We thus put it back with the other pieces, carefully respecting its rank, and try the next one, piece 4:
Still no luck. So we move forward to piece 5, and it fits in orientation b, with the backbone in front:
Now, looking for the piece that should go in position D, we start again from the bottom of our list, which is currently piece 3. It fits in orientation b, backbone up.
However, if we look carefully, we can see that we have completely blocked the way for any piece to fit in position E.
It proves that the configuration A1a, B2d, C5b, D3b is wrong.
We thus remove the last piece that was tried, which was piece 3 in position D, orientation b, and go on with the other possibilities. Next orientation? There is none, b was the last one. Thus piece 3 can't fit here at all.
We put it back and try the next one, which is piece 4. It seems to fit in orientation b, but...
How to get it in? We have there a problem: the configuration A1a, B2d, C5b, D4b is an impossible object!
Whatever assembly sequence we try, there is no way to assemble together the pieces 1, 2, 5 and 4 this way. This is a rare situation in 6-piece burrs, but it just means that our last try, D4b, was wrong.
So we move on to the next orientation for piece 4. It was the last one, so we try the next one, piece 6.
It doesn't fit, and there are no more pieces to try.
Conclusion, the configuration A1a, B2d, C5b is wrong. Nothing can fit in position D, in whatever orientation.
We thus remove the last piece of this configuration, which is the one in position C. It was piece 5, orientation b. No more orientation to try? We move to piece 6 for position C:
Then, in position D, piece 3 orientation b leads to the same problem as before: no room for anything in position E.
The next possibility that fits in D is piece 4 orientation b.
Then, in E, piece 3 fits in orientation a.
Finally, we can see that the last piece fits in the last position. We have found our first possible assembly, which is A1a, B2d, C6b, D4b, E3a, F5a. For the time being, we can't insert the last piece.
Looking at the possible movements, we quickly realize that if the last piece was in place, nothing at all could move. The puzzle would be impossible to disassemble!
Thus, A1a, B2d, C6b, D4b, E3a, F5a is a wrong configuration. So we remove the last piece, that was in position F, and try the next orientation: F5b. It doesn't fit. So we remove it. It was the last piece available, so we remove the previous one, E3a, and try the next orientation, E3b. It doesn't fit. The next piece is piece 5, and it fits in orientation b.
We are now in this configuration:
But the last piece doesn't fit at all. So the piece 5b in position E is wrong. We remove it...
It was the last piece in the last orientation, thus the previous one, D4b, is wrong too
The next piece to try in position D is piece 5, and it fits in position b.
Then piece 3 fits in E in orientation b,
And the last piece could fit in the last position, but again, it can't be inserted right now.
This time, though, the pieces are not blocked. The bottom one, A, can move backward...
Which allows a serial of other moves that, thanks to the simplicity of 6-piece burrs, we should be able to explore completely without too much difficulty.
The tricky part is to imagine that the piece F4a is in place. We are looking if the puzzle, assembled as such, can be disassembled. Therefore we need to perform only moves that could be allowed if it was in place. We also need to imagine how it could itself move:
This is a success! We can see that the puzzle would open in two halves, and that these halves are themselves disassemblable. Now, we just have to place piece F4a where it should be, and perform all the moves in the backwards direction:
We have found one solution. Maybe there are other ones. In order to find out, we would just need to remove the last piece, and go on trying the next possibilities.
If we proceed carefully, all possibilities will be eventually tried, and if the puzzle has a solution we should not miss it.
