# Strategy for solving tangram puzzles

It might seem at first, that there is no general strategy for solving tangram puzzles. You might imagine that the first piece could be placed at an infinite number of locations, rotated at infinitely many angles, and thus a general strategy would be very difficult to find. But there are a few things you will notice if you do enough of these:

## Looking at regions

The first thing to look for are regions where only one piece can fit. These are easy to identify, and placing these pieces first will drastically reduce the search space. In general, smaller regions have fewer possibilities for combinations of shapes to fill them, and so we should look to filling smaller regions before larger ones. It's also useful to consider the pieces that can be made by composing other pieces. For example, the square can be made up of two small triangles. The parallelogram can be made from two small triangles, plus a square in the middle. How is this useful? Imagine that you notice a region in the puzzle that must be filled by the square piece, or two triangular pieces. You also have the square and two triangular pieces available. In this case, you should pick the square piece to fill the region, as having two pieces, the small triangles, offers you greater mobility in solving the remainder of the puzzle.

## Looking at angles

Every vertex in a tangram puzzle must be the vertex of at least one piece. After noticing this, you will see that every interior angle in the puzzle must be filled by the interior angles of the pieces. Sometimes, an angle can be filled in only one way, or a small number of ways. It's also useful to note that every interior angle, of every piece, must be a \(45\) or \(90\) degree angle, because a tangram is made by cutting pieces from a square, using only horizontal, vertical, and diagonal cuts. Thus, every angle in a tangram must be a multiple of \(45^\circ,\) and further, angles greater than \(45^\circ\) can often be filled in multiple ways, depending on the pieces left, of course. This suggests that fewer possibilities, for placing the next piece, are more likely to occur at smaller angles, so we should look to those first.

## Looking at edges

Every edge in a tangram puzzle must consist entirely of edges from one or more pieces. Thus, for each edge, you can look for combinations of pieces to fill that edge. Further, notice that longer edges can often be filled in more ways than shorter edges, and thus, it's usually best to focus on the shorter edges first.

## Backtracking

Some puzzles will force you to try something and see where it leads, or look several moves ahead. I prefer placing pieces until I get stuck. Then I swap out a piece that's been placed with some other piece that's available. In this way, solving a tangram puzzle is very much like solving a maze. One strategy to solving a maze is to wander around until you hit a dead end, then backtrack until you find an alternate path, and try that. Of course, the difficulty with this strategy is remembering what you've tried, but since there are only 7 pieces in a tangram puzzle, and all puzzles must force at least one move, the number of possible configurations is low, and so remembering where you've been is not egregious.

## Applying the tactics without backtracking

Every time a piece is placed, new boundary edges form, making each of the aforementioned tactics relevant again. You, as the teacher, shouldn't tell students this strategy immediately, you should lead them to discover it. Now let's apply these tactics, to solve the "friend" puzzle below:

We notice the feet can only be filled by the smallest triangle, so we fill those regions first. Now the head can only be filled by the square. After that, the thigh of the planted leg must be the parallelogram, and the thigh of the leg that's lifted must be the mid-sized triangle. Each of the angles at the hands can only be filled in one way. After placing those two large triangles, the puzzle is complete.

## Applying the tactics with backtracking

As you can see, this was an easy puzzle to solve, because most regions throughout the puzzle, could only be filled in one way. Harder puzzles will have long edges where different combinations of shapes must be tried, some of which lead to an unsolvable state, or "dead end." Below is a somewhat harder puzzle, because the region, each angle, and each edge can be filled in multiple ways. Thus, you are forced to try something and see where it leads, or imagine several moves at once.

## Generalizing tangrams

One way we could make tangram puzzles more difficult, is to make puzzles that require multiple sets of tangrams. Another way is to cut up the square, or some other shape, into other pieces. For example, make we cut a shape so that 30, 60, 90 angles appear, instead of just 45 and 90 angles. Could a 3D variant be made? Each polyhedron could be rested on the ground, or on another polyhedron. Or we could place the pieces via computer program, so that gravity might be ignored.