Bridges for Sudoku Players: What Carries Over and What Is New

If you play sudoku, most of what you already know carries over to Bridges, also called Hashi or Hashiwokakero. You still count against fixed totals, you still ask "where can this still go?", and you still make progress by ruling things out rather than guessing. Two ideas are new: bridges take up space and cannot cross, and every island must end up joined in one group. This post is for sudoku players who are new to Bridges.

Both puzzles have a home at the same publisher. The Japanese publisher Nikoli gave sudoku its name in 1984, after finding the puzzle in an American magazine as Number Place (Nikoli: Sudoku), and Nikoli also publishes Hashiwokakero (Nikoli: Hashiwokakero). The name means roughly "build bridges!" (Wikipedia: Hashiwokakero). It is no surprise the two feel related: both are pure logic, with one answer and no maths beyond counting.

The rules of Bridges, for a sudoku player

Bridges is played on a grid of islands, each showing a number. You join the islands with bridges until every island has exactly as many bridges as its number. The rules, in plain words:

1. Count exactly. The number on an island is how many bridges touch it.
2. Straight lines only. A bridge runs horizontally or vertically between two islands, never diagonally and never round a corner.
3. At most two between a pair. Two islands can be joined by no bridge, one, or a double, but never three.
4. No crossing. A bridge cannot cross another bridge or pass over an island.
5. Everything joins. When you are done, all the islands form one connected group.

Here is how that compares with sudoku:

In Puzzle Picnic you build a bridge by tapping an island and then a neighbour; tap again for a double. The four levels grow from a 7x7 grid on Easy to 9x9 on Medium, 9x12 on Hard and 11x14 on Expert. For the full rules with a first puzzle, see how to play Bridges.

What carries over: counting to a fixed total

The most useful sudoku habit in Bridges is counting against a total that cannot change. In sudoku, a row must hold each digit once; in Bridges, an island must have exactly its number of bridges, and its neighbours put a ceiling on how many it can get.

Every island can only reach the nearest island in each of up to four directions, and each of those connections holds at most two bridges. So an island with n neighbours can take at most 2n bridges. Compare that maximum with the number:

The "one less than the maximum" rows deserve a sentence of proof, because they are the Bridges version of a sudoku elimination. Take a 3 in a corner with two neighbours. If it had no bridge to one of them, the other could give it at most 2, which is less than 3. So it must have at least one bridge to each.

The ceiling also shrinks as the puzzle goes on, just as candidates do in sudoku. A neighbour that is a 1 can give at most one bridge. A neighbour that is already full can give none. Counting with those real limits, rather than the starting ones, is where most Bridges progress comes from. The Hashi strategy guide has the full set of these counting rules.

What carries over: "where can it go?"

In sudoku you constantly ask where a digit can still go in a row or box. In Bridges, the same question is "where can this island's bridges still go?", and the answer is a short list of neighbours.

An island with one neighbour is the Bridges version of a cell with one candidate. A 2 with a single neighbour must have a double bridge to it, with nothing to think about. Like singles in sudoku, these are the first things to look for.

A full island is the Bridges version of a placed digit. In sudoku, once you place a 7 you remove 7 from every cell that sees it. In Bridges, once an island has all its bridges, it takes no more, so its neighbours lose it as an option. That often turns a neighbour with two options into a neighbour with one.

Sudoku players who keep notes will find the same thinking useful here, kept in your head instead of in the cells: for each island, "needs 2 more, two neighbours left, one of them can only take one". That sentence is a pencil mark. If you are a heavy note-taker in sudoku, sudoku notes and pencil marks explains why the habit pays off, and it pays off here for the same reason.

What is new: bridges take up space

The first idea with no sudoku equivalent is that bridges have a shape and get in each other's way. A sudoku digit only affects the cells that share its row, column or box. A Bridges bridge occupies every square between its two islands, and no other bridge may cross it.

		2
3	-	-	-	4
		1

That makes every bridge a deduction for its neighbours. In the fragment, the 2 at the top has just lost a neighbour. If it had only two neighbours before, it now has one left, and the "one neighbour" rule above may finish it.

It works the other way too. Suppose an island can only reach its number by sending a bridge across a gap, and a second bridge elsewhere would cross that gap. The first bridge is forced, so the crossing one is ruled out before you ever draw it. In sudoku, a placed digit never takes up room in another cell; in Bridges, always look along the line a new bridge blocks.

What is new: everything must join

The second new idea is connectivity: when the puzzle is finished, all the islands must form one connected group. Sudoku has no rule about the grid as a whole, so sudoku players often forget this one, and it is one of the most powerful in Bridges.

The rule turns into concrete deductions about islands that would close a group too early:

• Two 1s cannot be joined to each other. A 1 joined to a 1 makes a pair with no room for any other bridge, cut off from everything else. (The only exception is a puzzle with just those two islands.)
• Two 2s cannot be joined by a double. The double fills both islands and again leaves a closed pair.
• A group must keep a way out. More generally, any bridge that would finish off a group of islands, leaving none of them with bridges to spare, is ruled out, unless that group is the whole puzzle.

1

1

This is a different kind of reasoning from sudoku: you look at the grid as a map of groups rather than as cells. The post on the connectivity rule goes much further with it.

A first Bridges puzzle, solved like a sudoku

Here is a complete Bridges puzzle on a 7x7 grid with eight islands. Every move below has a sudoku twin, and a brute-force search confirms the puzzle has exactly one solution. Rows are numbered 1 to 7 from the top, columns 1 to 7 from the left.

			2
					2
1			5		5
2		3			2

Step 1: islands with one neighbour (the singles)

Look along each island's row and column for the nearest island. The 2 at the top (row 1, column 4) has only one neighbour: the 5 straight below it. So it needs a double bridge down to that 5. The 2 in row 3, column 6, also has just one neighbour, the 5 below it: another double.

Step 2: the 5 in the middle (a full house)

The 5 in row 5, column 4, has three neighbours: the 2 above, the 1 to its left and the 5 to its right. The most it can get is 2 from above, 1 from the 1 (a 1 can only give one) and 2 from the right: 2 + 1 + 2 = 5. It needs exactly 5, so every one of those is forced, like the last empty cell in a sudoku row. Build a single bridge to the 1 and a double to the 5 on the right.

Step 3: the 1 is full (elimination)

The 1 in row 5, column 1, now has its bridge. It takes no more, so the 2 below it cannot be joined to it. That 2 loses a neighbour, just as a sudoku cell loses a candidate when the digit is placed nearby.

Step 4: the 5 on the right (counting what is left)

The 5 in row 5, column 6, already has a double from above and a double from the left: 4 bridges. It needs one more, and its only other neighbour is the 2 below it. Build a single bridge down.

Step 5: down the bottom row

The 2 in row 7, column 6, has one bridge and needs one more; its only other neighbour is the 3 to its left. Build a single. The 3 now has one bridge and needs two more, and its only other neighbour is the 2 on the far left: a double. That fills the 2 on the left as well.

			2
			||
			||		2
			||		||
1	-	-	5	=	5
					|
2	=	3	-	-	2

Step 6: check that everything joins

The last check is the one sudoku does not have. Follow the bridges: the top 2, the 1 and both 5s are joined; the right 5 leads down to the bottom row; and the bottom row joins the 3 and the left 2. All eight islands form one group, so the puzzle is solved.

Where sudoku habits can mislead you

A few sudoku reflexes need adjusting in Bridges. They are worth knowing before your first harder grid.

• Looking only at one island at a time. In sudoku, a cell is affected only by its units. In Bridges, a bridge you build can block a crossing far from where you were looking. After each bridge, glance along it.
• Forgetting the group rule until the end. Sudoku players tend to check connectivity last, as in step 6. On harder puzzles, it is a source of deductions from the start: two 1s facing each other, or two 2s, already tell you something.
• Counting cells instead of bridges. The number on an island counts bridges, not neighbours. A 4 with two neighbours needs two doubles, not "four connections".
• Expecting notes to do the work. Sudoku has a natural place to write candidates. Bridges does not, so the counting happens in your head. Say it in words: "needs 2, two neighbours, one is full".

Most of the classic beginner slips in Bridges come from these four, and common Bridges mistakes shows how to spot and undo each one.

Frequently asked questions

Is Bridges like sudoku?

Yes, in the ways that matter most. Both are logic puzzles with exactly one solution, both are solved by counting against fixed totals and ruling options out, and neither needs maths or guessing. Bridges adds two ideas sudoku does not have: bridges cannot cross, and every island must join into one group.

Is Bridges harder than sudoku?

Not by nature. Both come in gentle and very hard forms: they get harder as grids grow and the techniques needed get deeper. Sudoku players usually find the counting in Bridges familiar, so the main thing to learn is the two new rules about crossing and joining.

What is the sudoku equivalent of a naked single in Bridges?

An island with only one neighbour. All its bridges must go to that neighbour, just as a cell with one candidate must take that digit. The next closest equivalent is an island whose number equals the most its neighbours can give, which works like a full house.

Do I need to guess in Bridges?

No. A well-made Bridges puzzle has exactly one solution, and there is always a next step you can prove. Every puzzle in Puzzle Picnic is checked by a solver to have exactly one solution, and a hint explains the next logical step if you are stuck.

Is Hashi the same as Bridges?

Yes. Hashiwokakero is Nikoli's name for the puzzle, Hashi is the short form, and Bridges is the English name. Some publishers have used other names too, such as Chopsticks and Ai-Ki-Ai.

Where to go next

If the puzzle above went smoothly, you already have the two habits that matter most: count each island's number against its neighbours, and treat full islands like placed digits. Next, read how to play Bridges for the rules in full, then the Hashi strategy guide for edge islands, crossings and the connectivity deductions that harder grids need. Sudoku players who want to keep both puzzles going will find the technique ladder for sudoku in sudoku strategies.

You can try a 7x7 Easy grid on the Bridges page, in the browser or in the app, and step up through the levels when the counting starts to feel automatic.

Sources

• Nikoli: Hashiwokakero rules
• Nikoli: Sudoku
• Wikipedia: Hashiwokakero