Reading Difficulty: 0/5
Prerequisites: Have a brain

For a nice change of pace from the mile-long posts, I've decided to split this topic into two short ones.

This post in particular won't have math in it! The next one will. Here, I will just introduce the Heyawake puzzle's rules. I'll give a sample puzzle and walk through how to solve it. Then you can try and solve some on your own!

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What is Heyawake?

Heyawake (probably pronounced "Hay-Ah-Wah-Keh") is a logic puzzle, like Sudoku. You're given a thing, and you need to fill out the thing in some logical way to satisfy a collection of rules.

Here is an example solve:

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What are the rules?

1. A grid full of "rooms" is given. Rooms are enclosed by the bold lines. Some rooms may contain a single number.
2. Fill some squares in black.
3. If a room has the number , then that room must have exactly black squares. Rooms that have no number can have any number of black squares.
4. Two block squares may NEVER be orthogonally adjacent! (They can touch diagonally, however).
5. There cannot exist a horizontal or vertical line that cuts through a room without hitting a black square (or the edge of the grid). This is the essential rule of Heyawake! I will give examples to clarify precisely what this means.
6. The white squares must form a single connected/contiguous region. (This is sometimes called the "connectivity" condition.)

Here are sample errors.

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Sample Solve

Let's solve the example puzzle together! If you want, here's a link to the puzzle if you want to follow along and/or try it yourself first: https://puzz.link/p?heyawake/7/7/94i94i94g01vg0fs0013102g2h.

Step 1

Step 2

Step 3

Step 4

Step 5

Step 6

Step 7

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A Historical Note on the "Essential Rule"

I phrased it in a more "intuitive" way, rather than trying to be precise. I think it is hard to describe precisely.

In the past, the essential rule would typically be phrased like this: Sounds legit, right? Why didn't I just say that? That's because it's technically wrong!

...But back in the day, it was correct. What changed?

Answer: Modern times introduced non-rectangular rooms. If the puzzle contained only rectangular rooms (as is true in many logic puzzle books), then the rule is indeed equivalent! Now, consider the following example:



Under the original wording, this red line of white squares is not a violation of the rules, because it technically passes through only two rooms. It exits and re-enters the same room. But this certainly feels very wrong because it "cuts through" a room, in the way that I have intuitively described it in the rules.

To fix this, some write the essential rule as follows: It fixes the rectangular issue, albeit there is still some imprecision (e.g. does ending at a border line count as an intersection?), which is why I chose a more intuitive wording in lieu of being precise.

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Puzzles For You!

Give it a try! Left click to shade in a black square, right click to mark a square as not black. You can also undo with Ctrl+Z.

Easy puzzle: https://puzz.link/p?heyawake/6/6/5dd0arsifop122j1g2h

Easy-Medium puzzle: https://puzz.link/p?heyawake/7/7/a5kqd6g801vg00dv002i324g3

Medium puzzle: https://puzz.link/p?heyawake/7/7/4i1052h2g081vo00c044g52g

Hard puzzle: https://puzz.link/p?heyawake/7/7/e75ipcpcgpv100tv00h31i111121 (Preliminary hint: This may require both creativity and some casework! Use the "trial" mode to easily explore possible paths.) Big Hint

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For Next Time...

Great, now try solving this one.



https://puzz.link/p?heyawake/10/10/00000112244000000000003s00003s000000-1f2

...You can probably see that it's very hard, and there doesn't seem to be any good way to solve it.

In the next post, we will explore how we can solve puzzles like the above... using math!