@tuna
@discuss.tchncs.deIn the post explaining the hint "one specific dot in all solutions imaginable" I cut the post short saying I had 2 strategies to help find the dots quicker. Well, it's because I realized my existing strategy might not be the most optimized. I've finally figured out how to combine both strategies into one.
Let's jump straight into the algorithm, using the same board as the previous post.
⚫️⚫️⚫️⚫️⚫️
⚫️⚫️⚫️⚫️⚫️
⚫️9️⃣⚫️⚫️⚫️
⚫️⚫️⚫️⚫️⚫️
⚫️⚫️⚫️⚫️⚫️
⚫️⚫️⚫️⚫️⚫️
⚫️⚫️⚫️⚫️⚫️
Choose 3 of the shortest directions, and count upward. The 4th axis contains the "overflow"
Here's how I count up in my head:
⚫️3️⃣⚫️⚫️⚫️
⚫️2️⃣⚫️⚫️⚫️
1️⃣#️⃣4️⃣5️⃣6️⃣
⚫️⚫️⚫️⚫️⚫️
⚫️⚫️⚫️⚫️⚫️
⚫️⚫️⚫️⚫️⚫️
⚫️⚫️⚫️⚫️⚫️
Fill in the remaining dots blue
⚫️⚫️⚫️⚫️⚫️
⚫️⚫️⚫️⚫️⚫️
⚫️#️⃣⚫️⚫️⚫️
⚫️7️⃣⚫️⚫️⚫️
⚫️8️⃣⚫️⚫️⚫️
⚫️9️⃣⚫️⚫️⚫️
⚫️⚫️⚫️⚫️⚫️
⚫️⚫️⚫️⚫️⚫️
⚫️⚫️⚫️⚫️⚫️
⚫️9️⃣⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
⚫️⚫️⚫️⚫️⚫️
Start from 0 and count how many dots away from the nearest wall, in the same direction. I'll call the number of dots between the last blue dot and the first wall the "blue gap." In this case, it is 1.
⚫️⚫️⚫️⚫️⚫️
⚫️⚫️⚫️⚫️⚫️
⚫️9️⃣⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
⚫️1️⃣⚫️⚫️⚫️
Go to the nearest wall in the other 3 directions, and apply the same blue gap. The rest of the dots in that direction are blue
⚫️1️⃣⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
1️⃣9️⃣🔵🔵1️⃣
⚫️🔵⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
⚫️⚫️⚫️⚫️⚫️
Tada:
⚫️⚫️⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
⚫️9️⃣🔵🔵⚫️
⚫️🔵⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
⚫️⚫️⚫️⚫️⚫️
I learned about the blue gap today. I might be able to get some new PBs with this strat!
This is my favorite strategy, because it always makes me feel amazing when I spot it!
I'm hoping to explain the blind dot rule by taking you along a similar journey I took to discover the blind dot rule. Enjoy :)
In a time before I came to fully understand what the blind dot rule was, I started to pick up on these areas that looked like little crevices (this is what we will call them for now). There was some sort of pattern going on, but I couldn't put my finger on what it was.
Here's a 5x5 board I got one day, full of these crevices:
⚫️2️⃣🔴⚫️🔴
⚫️⚫️4️⃣⚫️⚫️
⚫️⚫️3️⃣⚫️⚫️
3️⃣🔴3️⃣⚫️⚫️
🔴⚫️⚫️🔴⚫️
Let's solve step-by-step, and listen to what these crevices (denoted by 🔳) tell us:
⚫️2️⃣🔴🔳🔴
⚫️⚫️4️⃣⚫️⚫️
⚫️⚫️3️⃣⚫️⚫️
3️⃣🔴3️⃣⚫️⚫️
🔴🔳⚫️🔴🔳
🔵2️⃣🔴⚫️🔴
🔵⚫️4️⃣⚫️⚫️
🔵⚫️3️⃣⚫️⚫️
3️⃣🔴3️⃣⚫️⚫️
🔴⚫️⚫️🔴⚫️
🔵2️⃣🔴⚫️🔴
🔵🔵4️⃣⚫️⚫️
🔵🔴3️⃣⚫️⚫️
3️⃣🔴3️⃣⚫️⚫️
🔴⚫️⚫️🔴⚫️
The 4️⃣ can see all its dots.
🔵2️⃣🔴⚫️🔴
🔵🔵4️⃣🔴⚫️
🔵🔴3️⃣⚫️⚫️
3️⃣🔴3️⃣⚫️⚫️
🔴⚫️🔴🔴⚫️
🔵2️⃣🔴🔴🔴
🔵🔵4️⃣🔴⚫️
🔵🔴3️⃣⚫️⚫️
3️⃣🔴3️⃣⚫️⚫️
🔴🔴🔴🔴⚫️
Alright, pause.
It seems that those 2 crevices became red! There was also a new crevice made in the process. Maybe those will turn red too?
🔵2️⃣🔴🔴🔴
🔵🔵4️⃣🔴🔳
🔵🔴3️⃣⚫️⚫️
3️⃣🔴3️⃣⚫️⚫️
🔴🔴🔴🔴🔳
Let's continue solving:
🔵2️⃣🔴🔴🔴
🔵🔵4️⃣🔴⚫️
🔵🔴3️⃣🔵⚫️
3️⃣🔴3️⃣🔵⚫️
🔴🔴🔴🔴⚫️
Both 3️⃣s can see all their dots.
🔵2️⃣🔴🔴🔴
🔵🔵4️⃣🔴⚫️
🔵🔴3️⃣🔵🔴
3️⃣🔴3️⃣🔵🔴
🔴🔴🔴🔴⚫️
🔵2️⃣🔴🔴🔴
🔵🔵4️⃣🔴🔴
🔵🔴3️⃣🔵🔴
3️⃣🔴3️⃣🔵🔴
🔴🔴🔴🔴🔴
If you paid especially close attention, you might've noticed that not only did all the crevices become red, every "entrance" to their cove became red too.
One explanation involves the property explained in a previous post, that a board has one, and only one solution.
Suppose we had a board like this instead:
🔵2️⃣🔴🔴🔴
🔵🔵⚫️🔴⚫️
🔵🔴3️⃣🔵⚫️
3️⃣🔴3️⃣🔵⚫️
🔴🔴🔴🔴⚫️
Let's assume that they are blues:
🔵2️⃣🔴🔴🔴
🔵🔵⚫️🔴⚫️
🔵🔴3️⃣🔵🔵
3️⃣🔴3️⃣🔵🔵
🔴🔴🔴🔴⚫️
Then the crevices would be ambiguous, or allow the board to contain more than 1 solution:
🔵2️⃣🔴🔴🔴
🔵🔵🔴🔴❔
🔵🔴3️⃣🔵🔵
3️⃣🔴3️⃣🔵🔵
🔴🔴🔴🔴❔
So, if it can't be blue, then it must be red!
🔵2️⃣🔴🔴🔴
🔵🔵🔵🔴⚫️
🔵🔴3️⃣🔵🔴
3️⃣🔴3️⃣🔵🔴
🔴🔴🔴🔴⚫️
Let's use our new understanding to find and solve other boards using this type of pattern.
...hey look!
5️⃣⚫️⚫️5️⃣⚫️
⚫️⚫️4️⃣⚫️3️⃣
⚫️⚫️⚫️⚫️3️⃣
⚫️⚫️⚫️🔴⚫️
⚫️🔴🔴🔴🔴
What about this one? Something seems special around here:
5️⃣⚫️⚫️5️⃣⚫️
⚫️⚫️4️⃣⚫️3️⃣
⚫️⚫️⚫️⚫️3️⃣
⚫️🔳⚫️🔴⚫️
⚫️🔴🔴🔴🔴
If you imagine each blue dot is like a 4-way laser, it would cover the entire board except for that spot!
5️⃣⚫️⚫️5️⃣⚫️
⬇️⚫️4️⃣⚫️3️⃣
⬅️⬅️⬅️⬅️3️⃣
⬇️😎⬇️🔴⚫️
⬇️🔴🔴🔴🔴
In a way, it's kind of like the crevices. They also don't have any blue dots looking at them. See?
⚫️2️⃣🔴😎🔴
⚫️⚫️4️⃣➡️➡️
⚫️⚫️3️⃣⚫️⚫️
3️⃣🔴3️⃣➡️➡️
🔴😎⬇️🔴😎
Let's solve it and see what happens. Make a prediction!
::: spoiler Click to show solving steps
5️⃣⚫️⚫️5️⃣⚫️
⚫️⚫️4️⃣🔵3️⃣
⚫️⚫️⚫️⚫️3️⃣
⚫️⚫️⚫️🔴⚫️
⚫️🔴🔴🔴🔴
5️⃣⚫️⚫️5️⃣🔴
⚫️🔴4️⃣🔵3️⃣
⚫️⚫️⚫️⚫️3️⃣
⚫️⚫️⚫️🔴🔴
⚫️🔴🔴🔴🔴
5️⃣⚫️⚫️5️⃣🔴
⚫️🔴4️⃣🔵3️⃣
⚫️⚫️🔵🔵3️⃣
⚫️⚫️⚫️🔴🔴
⚫️🔴🔴🔴🔴
5️⃣⚫️⚫️5️⃣🔴
⚫️🔴4️⃣🔵3️⃣
⚫️🔴🔵🔵3️⃣
⚫️⚫️⚫️🔴🔴
⚫️🔴🔴🔴🔴
5️⃣🔵🔵5️⃣🔴
⚫️🔴4️⃣🔵3️⃣
⚫️🔴🔵🔵3️⃣
⚫️⚫️⚫️🔴🔴
⚫️🔴🔴🔴🔴
5️⃣🔵🔵5️⃣🔴
🔵🔴4️⃣🔵3️⃣
🔵🔴🔵🔵3️⃣
⚫️⚫️⚫️🔴🔴
⚫️🔴🔴🔴🔴
5️⃣🔵🔵5️⃣🔴
🔵🔴4️⃣🔵3️⃣
🔵🔴🔵🔵3️⃣
🔴⚫️⚫️🔴🔴
🔴🔴🔴🔴🔴
5️⃣🔵🔵5️⃣🔴
🔵🔴4️⃣🔵3️⃣
🔵🔴🔵🔵3️⃣
🔴⚫️🔴🔴🔴
🔴🔴🔴🔴🔴
:::
5️⃣🔵🔵5️⃣🔴
🔵🔴4️⃣🔵3️⃣
🔵🔴🔵🔵3️⃣
🔴🔴🔴🔴🔴
🔴🔴🔴🔴🔴
Cool! The dot indeed ended up red! Not only that, but ALL 3 of its neighbors ended up red as well!
It seems like the pattern has less to do with crevices, and more to do with whether any numbered blue dot can see a given dot.
Let's call the dots that can't be seen blind dots. For the dots immediately adjacent to the blind dot, let's call them guard dots.
Let's go over one final example for this post. In the previous examples, we assumed the line of sight of numbered blue dots extends arbitrarily outward, but I want to show that there is technically a limit.
Let's demonstrate by example, using this board:
🔴⚫️⚫️⚫️1️⃣
🔴🔴2️⃣⚫️⚫️
🔴1️⃣⚫️⚫️2️⃣
⚫️⚫️⚫️2️⃣⚫️
🔴⚫️⚫️3️⃣⚫️
Here are the areas I'm claiming to be blind dots:
🔴🔳⚫️⚫️1️⃣
🔴🔴2️⃣⚫️⚫️
🔴1️⃣⚫️⚫️2️⃣
🔳⚫️⚫️2️⃣⚫️
🔴⚫️⚫️3️⃣⚫️
You might think these can't be blind dots, because the numbered dots has line of sight to them:
🔴😧⬅️⬅️#️⃣
🔴🔴⚫️⚫️⚫️
🔴⚫️⚫️⚫️⚫️
😧⬅️⬅️#️⃣⚫️
🔴⚫️⚫️⚫️⚫️
But they don't have line of sight, because the 1️⃣ in the top right can only see one dot to the west, and likewise, the 2️⃣ near the bottom can only see one more dot to the west before it would see all of its dots.
🔴😎❌🔵1️⃣
🔴🔴⚫️⚫️⚫️
🔴⚫️⚫️⚫️⚫️
😎❌🔵2️⃣⚫️
🔴⚫️⚫️3️⃣⚫️
Of course, let's solve it to double check!
::: spoiler Solution steps
🔴⚫️⚫️⚫️1️⃣
🔴🔴2️⃣⚫️⚫️
🔴1️⃣⚫️⚫️2️⃣
⚫️⚫️⚫️2️⃣⚫️
🔴⚫️⚫️3️⃣⚫️
Looking further south would exceed the 1️⃣:
🔴⚫️🔴🔵1️⃣
🔴🔴2️⃣⚫️🔴
🔴1️⃣⚫️⚫️2️⃣
⚫️⚫️⚫️2️⃣⚫️
🔴⚫️⚫️3️⃣⚫️
One specific dot included in all solutions imaginable for the 2️⃣:
🔴⚫️🔴🔵1️⃣
🔴🔴2️⃣⚫️🔴
🔴1️⃣🔵⚫️2️⃣
⚫️⚫️⚫️2️⃣⚫️
🔴⚫️⚫️3️⃣⚫️
The 1️⃣ can see all its dots:
🔴⚫️🔴🔵1️⃣
🔴🔴2️⃣⚫️🔴
🔴1️⃣🔵🔴2️⃣
⚫️🔴⚫️2️⃣⚫️
🔴⚫️⚫️3️⃣⚫️
Only one direction left to look in for the 2️⃣:
🔴⚫️🔴🔵1️⃣
🔴🔴2️⃣⚫️🔴
🔴1️⃣🔵🔴2️⃣
⚫️🔴⚫️2️⃣🔵
🔴⚫️⚫️3️⃣🔵
The 2️⃣ near the bottom can see all its dots:
🔴⚫️🔴🔵1️⃣
🔴🔴2️⃣⚫️🔴
🔴1️⃣🔵🔴2️⃣
⚫️🔴🔴2️⃣🔵
🔴⚫️⚫️3️⃣🔵
Only one direction left to look in for the 2️⃣ and 3️⃣:
🔴⚫️🔴🔵1️⃣
🔴🔴2️⃣🔵🔴
🔴1️⃣🔵🔴2️⃣
⚫️🔴🔴2️⃣🔵
🔴⚫️🔵3️⃣🔵
The 3️⃣ can see all its dots:
🔴⚫️🔴🔵1️⃣
🔴🔴2️⃣🔵🔴
🔴1️⃣🔵🔴2️⃣
⚫️🔴🔴2️⃣🔵
🔴🔴🔵3️⃣🔵
:::
Yep! Those are blind dots!
🔴🔴🔴🔵1️⃣
🔴🔴2️⃣🔵🔴
🔴1️⃣🔵🔴2️⃣
🔴🔴🔴2️⃣🔵
🔴🔴🔵3️⃣🔵
In summary, we learned two things:
The blind dot rule is helpful because it can place red dots in a way that constrains numbered blue dots, which makes it quicker to count.
In the next part, I'll explain the deal with the guard dots. It turns out, there is more to the blind dot rule than meets the eye ;)
See you in the next one!
When I first started out, this was a confusing hint. It is saying something specific, but at the same time it feels vague.
It's best to understand this hint with an example.
Let's suppose there exists a 5x7 board, with a 9️⃣ in this position:
⚫️⚫️⚫️⚫️⚫️
⚫️⚫️⚫️⚫️⚫️
⚫️9️⃣⚫️⚫️⚫️
⚫️⚫️⚫️⚫️⚫️
⚫️⚫️⚫️⚫️⚫️
⚫️⚫️⚫️⚫️⚫️
⚫️⚫️⚫️⚫️⚫️
Let's try coming up with valid solutions using trial-and-error, and see if we can spot any patterns:
⚫️🔵⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
🔵9️⃣🔵🔵🔵
⚫️🔵⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
⚫️🔴⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
🔴9️⃣🔵🔵🔵
⚫️🔵⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
🔵9️⃣🔵🔵🔴
⚫️🔵⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
⚫️🔴⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
🔵9️⃣🔵🔵🔵
⚫️🔵⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
It seems like no matter how we draw it, we always have blue dots in these positions:
⚫️⚫️⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
⚫️9️⃣🔵🔵⚫️
⚫️🔵⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
⚫️🔵⚫️⚫️⚫️
⚫️⚫️⚫️⚫️⚫️
That's what the hint means :)
There are 2 strategies I know of that can find where the blue dots are without having to exhaustively list all solutions. I'll have to share in future posts!
My 5x5 is pretty optimized, and the rest are decent. My 9x9 has some room for improvement though.
If you haven't played before and you'd like to play, you can do so here: https://0hn0.com
Just know that my first 9x9 took maybe 30+ minutes. It takes many, many boards to get competitive!
The fundamental rule is that board generation has one, and only one solution. It doesn't sound that insightful on the surface, but it is that assumption which led me to discover important advanced strategies, as well as spot other smaller, esoteric cases. This post covers one of those esoteric cases, but it illustrates the rule well.
Let me explain with an example, of a partially completed board:
🔵5️⃣4️⃣4️⃣🔵
⚫️🔵🔴🔴⚫️
⚫️🔴⚫️⚫️4️⃣
2️⃣2️⃣⚫️⚫️3️⃣
🔴⚫️⚫️5️⃣⚫️
Specifically, let's focus on the leftmost 2️⃣, and ignore the irrelevant parts:
🔵5️⃣🔵🔵🔵
⚫️🔵🔴⚫️⚫️
⚫️🔴⚫️⚫️⚫️
2️⃣⚫️⚫️⚫️⚫️
🔴⚫️⚫️⚫️⚫️
I am reasoning that it should be filled in like so:
🔵5️⃣🔵🔵🔵
🔴🔵🔴⚫️⚫️
🔵🔴⚫️⚫️⚫️
2️⃣🔵🔴⚫️⚫️
🔴⚫️⚫️⚫️⚫️
How can I be so sure? Suppose immediately above the 2️⃣ lives a red wall. The dot above that could be either red or blue (denoted by ❔)
🔵5️⃣🔵🔵🔵
❔🔵🔴⚫️⚫️
🔴🔴⚫️⚫️⚫️
2️⃣⚫️⚫️⚫️⚫️
🔴⚫️⚫️⚫️⚫️
In other words, the board would have two solutions, which breaks the rule that boards have a single solution.
So it must be a blue dot above the 2️⃣.
🔵5️⃣🔵🔵🔵
⚫️🔵🔴⚫️⚫️
🔵🔴⚫️⚫️⚫️
2️⃣⚫️⚫️⚫️⚫️
🔴⚫️⚫️⚫️⚫️
From here, conventional rules can be applied. Looking further north would exceed the 2️⃣, so it must be a red wall.
🔵5️⃣🔵🔵🔵
🔴🔵🔴⚫️⚫️
🔵🔴⚫️⚫️⚫️
2️⃣⚫️⚫️⚫️⚫️
🔴⚫️⚫️⚫️⚫️
And the 2️⃣ has only one direction left to look in.
🔵5️⃣🔵🔵🔵
🔴🔵🔴⚫️⚫️
🔵🔴⚫️⚫️⚫️
2️⃣🔵🔴⚫️⚫️
🔴⚫️⚫️⚫️⚫️
Let's add back the other dots and solve it, to see if it works!
🔵5️⃣4️⃣4️⃣🔵
🔴🔵🔴🔴⚫️
🔵🔴⚫️⚫️4️⃣
2️⃣2️⃣🔴⚫️3️⃣
🔴⚫️⚫️5️⃣⚫️
(Filling in the 2️⃣ and 5️⃣)
🔵5️⃣4️⃣4️⃣🔵
🔴🔵🔴🔴⚫️
🔵🔴⚫️🔵4️⃣
2️⃣2️⃣🔴🔵3️⃣
🔴🔵🔵5️⃣🔵
(3️⃣ can see all dots)
🔵5️⃣4️⃣4️⃣🔵
🔴🔵🔴🔴🔴
🔵🔴⚫️🔵4️⃣
2️⃣2️⃣🔴🔵3️⃣
🔴🔵🔵5️⃣🔵
(Only one direction left for 4️⃣ to look in)
🔵5️⃣4️⃣4️⃣🔵
🔴🔵🔴🔴🔴
🔵🔴🔵🔵4️⃣
2️⃣2️⃣🔴🔵3️⃣
🔴🔵🔵5️⃣🔵
Yay! It seems like this idea might help complete more boards.
I recommend using emojis to share boards when demonstrating hypotheticals.
I also sometimes use ▪️ to represent areas where numbered dots can't see.
When writing a board, add 4 spaces after each line to make it appear more compact. Here, the .
's represent spaces
⚫️⚫️⚫️2️⃣⚫️....
⚫️1️⃣⚫️⚫️2️⃣....
4️⃣🔴⚫️⚫️⚫️....
⚫️🔴⚫️⚫️⚫️....
⚫️🔴4️⃣4️⃣⚫️
Result:
⚫️⚫️⚫️2️⃣⚫️
⚫️1️⃣⚫️⚫️2️⃣
4️⃣🔴⚫️⚫️⚫️
⚫️🔴⚫️⚫️⚫️
⚫️🔴4️⃣4️⃣⚫️