The ROUNDABOUTS puzzle is one of the easiest to solve. There is really only one fundamental strategy, but there are a number of simple extensions
which are explained in the following.
When specific cells need to be referred to in this discussion, they will be designated as C4-R3 which means the 4th cell in row 3.
Fundamental Strategy.
This strategy requires you to look out for cells which are blocked in two directions, from which it follows that the path must enter and leave
the cell by means of the two directions which are not blocked. Initially, the only cells which meet this criteria are the four corner cells.
See the right hand graphic.
Further Steps.
When a path enters a roundabout, it must execute a 90 degree turn before it continues. This applies in C2-R1 and C6-R5.
When a path executes a 90 degree turn outside of a roundabout, it must continue in a straight line until it encounters a roundabout. This applies
in C1-R2, C5-R6 and C4-R6. Note also that in each case, it must then execute a 90 degree turn.
When a path leaves a roundabout, it must execute a 90 degree turn before it enters another roundabout. This applies to C2-R6 and C5-R1.
All of these changes are shown in the right hand graphic above.
More Steps.
The activity so far has exposed a number of cells in which the Fundamental Strategy can be applied. These are C3-R1, C4-R1, C1-R4, C6-R4.
Consider C2-R2. It might appear that there are two choices when continuing this path. However, continuing into C2-R3 is illegal, as this creates
a premature loop. The same argument applies to C6-R2.
These changes are shown in the right hand graphic above. Solvers should have no problem completing the puzzle from this point.
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