Bonus challenge offered to CS106B students winter 2012. Was the only student to solve it.
This program determines how to reassemble a glass window broken into triangular shards by dropping them in from above, one at a time. The user has the option of loading a puzzle from a saved file, or creating his own by drawing triangles on a grid. Once a puzzle is loaded or created, the user can then watch an animation of the triangles being dropped from above in the correct order.
- superheroes.cpp
To solve the puzzle, the program first looks through all the triangles to find the "starting triangles", or the ones that lie on the bottom side of the rectangular border and that are concave, meaning that if they are inserted now, they will not block any triangles later on. (It will be described later how to determine if a triangle will block another triangle.)
Below is an example of two valid starting triangles. They both lie on the bottom face and will not block any triangles later on.
______________________
| |
| |
| /\ |
| /\ / \ |
|___/__\_______/____\__| FIGURE 1
Below is an example of two invalid starting triangles. The first does not lie on the bottom face and the second will block a triangle later on (the one that touches its ride side).
______________________
| _// |
| /\ _/ / |
| /__\ _/ / |
| _/ / X |
|__________/___ /______| FIGURE 2
The program then stores a list of all the exposed, upward-facing sides of these starting triangles. The lines in this list are called boundary lines because they describe the boundary of all the triangles inserted so far.
There are four boundary lines in the example below (lines touching the border are not included).
______________________
| |
| |
| /\ |
| 1 /\ 2 3 / \ 4 |
|___/__\_______/____\__| FIGURE 3
Next, the program recursively searches through all the unused triangles until it finds one with a side that is also a boundary line. This triangle is inserted next if it will not block a triangle later on. (It will be described later how to determine if a triangle will block another triangle.) The boundary lines list is updated so that the contact line is removed (since it is no longer exposed), and any new, exposed, upward-facing sides of the newly inserted triangle are added to the boundary lines list.
In the example below, first there are three boundary lines. Once the new triangle is inserted, boundary line #1 is removed since it is a line of contact, and boundary line #4 is added since it is exposed and upward-facing.
______________________ ______________________
| | | |
| | | |
| | _______ |_4_____ |
| _/\ /| | _/ | _/\ /|
| 1 _/ \ 2 3 / | + | _/ = | _/ \ 2 3 / |
| _/ \ / | | _/ | _/ \ / |
|/_________\_______/___| |/ |/_________\_______/___| FIGURE 4
The list of boundary lines is continually updated so it always describes the exposed boundary of all the triangles inserted so far, as depicted in the example below.
______________________ ______________________
| | | |
| | | |
|_4_____ | |_4_____ |
| _/\ /| \\__ | _/\\__ /|
| _/ \ 2 3 / | + \ \__ = | _/ \ \__5 3 / |
| _/ \ / | \ \__ | _/ \ \__ / |
|/_________\_______/___| \______\ |/_________\______\/___| FIGURE 5
This recursive process is repeated until the list of boundary lines is reduced to a size of zero, at which point the puzzle is solved. There are no boundary lines in the example below (recal that lines on the rectangular border are not considered boundary lines).
______________________
| /\ __/ \__ |
| / \ __/ \__ |
|/____\_/_____________\|
| /\\__ /|
| / \ \__ / |
| / \ \__ / |
|/_________\______\/___| FIGURE 6
It was mentioned earlier that a new triangle is inserted only if one (or more) of its sides is a boundary line, and if inserting the triangle now will not block any triangles later on. How is it determined if inserting the triangle now will not block any triangles later on? Look at the point of the triangle that is not an endpoint of the contact line. In the examples below, the contact lines are between points A and B, and the third points are points C.
___________________________
| |
| |
|_______ A | A _______ C
| /\\__ | \__ \
| / \ \__ | + \__ \
| / \ \__ | \__\
|/_________\______\_B______| \\ B FIGURE 7
__________________________
| |
| |
|_______ A | A ______________ C
| /\\__ | \__ /
| / \ \__ | + \__ /
| / \ \__ | \__ /
|/_________\______\_B______| \/ B FIGURE 8
Is the x-coordinate of this third point (C) between the x-coordinates of the contact line endpoints (A and B)? In FIGURE 7, C is between A and B. This means that the triangle is safe to add now. In FIGURE 8, C is not between A and B. This means further tests need to be conducted to determine if the triangle is safe to add now, because this triangle might block another triangle with a side between B and C.
__________________________ __________________________
| | | |
| | | |
|_______ | ______________ C |______________________ |
| /\\__ | \__ / | /\\__ / |
| / \ \__ | + \__ / = | / \ \__ / |
| / \ \__ | \__ / | / \ \__ / X |
|/_________\______\________| \/ B |/_________\______\/_______| FIGURE 9
A triangle with an over-extending third point (C) will not bock any triangles later on if the line between the third point (C) and the lower contact line endpoint (B) is already a boundary line (since this means the triangle which would have been blocked has already been added). In the example below, the line between C and B is already a boundary line (4), so even though the x-coordinate of C is not between the x-coordinates of A and B, it is safe to add the triangle now.
__________________________ __________________________
| | | |
| | | |
|___1___ | A ______________ C |___1___________6______ |
| /\\__ /\ | \__ / | /\\__ /\ |
| / \ \__2 4 / \ 5| + \__2 / = | / \ \__ / \ 5|
| / \ \__ / \ | \__ / | / \ \__ / \ |
|/_________\______\/______\| \/ B |/_________\______\/______\| FIGURE 10
That's it! This algorithm will work for any combination of triangles, no matter how complicated. Try the pre-loaded puzzles, or create your own!