How to Solve Train Tracks Puzzles
Eight techniques, in the order our grading engine uses them. Easy puzzles need only the first six; medium puzzles add the seventh; hard puzzles may need the eighth.
| Technique | Needed from |
|---|---|
| 1-4. Counting, two connections, dead ends | Easy |
| 5-6. No loops, no early finish | Easy |
| 7. Crossing parity | Medium |
| 8. The “what if” test | Hard |
In the diagrams, black track is already known, orange shows what the technique proves, red shows a move the rule forbids, and orange dots are squares that must hold track in a direction still to be found.
1.Zeros and full lines
A 0 beside a row or under a column means the track never enters that line, so every square in it is empty. At the other extreme, a number equal to the width of the grid means every square in that line is track. Mark both kinds straight away: they are free information, and the squares they fix change the arithmetic for every line that crosses them.
2.Count what is left in each line
For every row and column keep two numbers in mind: track squares already fixed, and squares still undecided. If the fixed squares already equal the clue, the undecided ones are empty. If fixed plus undecided equals the clue, every undecided square is track. Most of the progress in any puzzle comes from repeating this check after each new mark.
3.Every track square has exactly two connections
A piece always joins exactly two of its four sides. So when a square is known to hold track and only two of its sides are still open (the others face an X, the grid edge, or a neighbour that is already full), the piece must use those two sides. The same rule works the other way: a square that already has two connections cannot take a third.
4.Dead ends and the grid edge
The track only leaves the grid at A and B. Any other square on the border has one fewer way out, and a corner square has two fewer. If a square has fewer than two sides that could still connect, it can never hold track, so mark it with an X. One new X often creates the next dead end beside it, so these tend to come in chains along the edges.
5.No loops
As you solve, the track appears as separate fragments. The two ends of the same fragment must never be joined, because that would close a loop that never reaches A or B. When both ends of one fragment sit next to the same square, that square cannot connect to both of them, which often decides which way it turns.
6.Don’t finish early
The fragment that starts at A and the fragment that ends at B may only be joined as the very last move. If joining them now would leave any row or column short of its number, that connection is impossible, and the ends must find another way.
7.Crossing parity
Draw an imaginary line straight across the grid between two rows (or down between two columns). Each time the track crosses it, the track switches sides. So if A and B are on opposite sides of the line, the track crosses it an odd number of times; if they are on the same side, an even number of times.
This matters when the counts allow only a few crossings. Suppose A and B are on opposite sides and just one column could still carry the track across. Then it must cross there, and both squares next to the line in that column are track. Or suppose they are on the same side and exactly one crossing is already fixed: a second crossing is now required, which often has only one possible place.
8.The “what if” test
When nothing else moves, pick one undecided square or connection and ask: what if it were empty (or used)? Apply techniques 1-7 to the consequences. If you reach a contradiction, such as a row with too much track, a dead end that must be track, or a loop, the assumption was wrong and the opposite is proven.
Choose the test carefully. The best candidates are squares with only two possibilities, next to a line that is nearly full or nearly empty, because contradictions show up within a few moves. Our hard puzzles never need more than one level of “what if”: you will not have to test an assumption inside another assumption.
Putting it together
A good rhythm is: count every line, mark what the counts force, follow each loose end as far as it goes, then count again. Only when a full pass finds nothing new should you look for a parity argument, and only after that for a “what if”. Practise the rhythm on easy puzzles, then test it on the daily puzzle. If you are stuck on a puzzle from a newspaper or book, the solver will check whether it has a unique answer.
Strategy questions
Which technique should I try first?
Always start by counting: zeros, full lines and lines where the remaining squares exactly match the number. They are quick and they feed every other technique.
Is using a “what if” test the same as guessing?
No. A guess commits to a choice and hopes. A “what if” test follows the forced consequences of a choice until they break a rule, then proves the opposite. You never keep a move you have not proven.
Why does parity work?
Every time the track crosses a straight line across the grid it switches sides. Starting on A’s side and finishing on B’s side takes an odd number of switches if they are on different sides and an even number if they are on the same side.