2021 Unpainted Masyu

Question

Paint some white circles to black such that the Masyu will be uniquely solvable and:

• Exactly 2 of 11 white circles on the left forming '2' should be painted black.
• Exactly 0 of 12 white circles on the middle-left forming '0' should be painted black.
• Exactly 2 of 11 white circles on the middle-right forming '2' should be painted black.
• Exactly 1 of 11 white circles on the right forming '1' should be painted black.

Answer 1

Solution

Step 1

We start with some simple deductions based on the locations of the shaded circles and the fact that the 0 has no shaded circles

Step 2

In the first '2', neither the bottom left circle nor its adjacent can be shaded and the path must pass through these horizontally. This means that the next circle along must be shaded. Also notice that, since the '1' has only 1 shaded circle it must be in the bottom half, i.e, the top 5 circles are all unshaded. With these deductions, we have the following

Step 3

The path cannot go through the bottom circles of the '0' horizontally (otherwise we generate a small closed loop down here). This forces the path through the shaded circle below this to go right. Since the bottom left circle of the second '2' is now in a corner, it must be shaded, as must either the 3rd or 4th circle in that row. Hence all other circles in the second '2' will be unshaded. After using these deductions, we get the following

Step 4

Where the path passes vertically through the middle of the first '2', the circle must be unshaded. If the circle diagonally above this is shaded, it forces the next circle diagonally above to also be shaded (which results in too many for the '2'). Staying with this circle, if the path goes through it vertically, then it forces the circle two above into a corner and for it to be shaded. This would mean that all other circles in the first '2' would be unshaded. After a few simple deductions we arrive at the following situation.
We can see that there is no way to join up the ends near the bottom left. Hence, the path must pass horizontally through this unshaded circle in the first '2'. If the circle diagonally above this is shaded, we also arrive at an untenable situation after a few deductions.

Hence, this circle must be unshaded. If the circle at the bottom right of the first '2' is shaded this leads to a closed loop at the bottom and it must also be unshaded. A few more straight forward Masyu deductions lead us to the following situation.

Step 5

We now see that the last circle in the bottom row of the second '2' must be the one to be shaded. Also, in the first '2' it's not hard to deduce that only the top right of the remaining undefined circles can be shaded. This allows us to complete the grid on that side. A few more standard Masyu deductions lead us to the following

Step 6

Looking at the bottom row of the '1' case by case, we can quickly see that none of the circles here can be shaded (the ends lead to three unshaded circles begin crossed horizontally, the second leads to an unshaded circle in a corner and the third forces the path to intersect when travelling vertically). Hence all are unshaded and the path passes vertically through all of them. Because of this, the circle directly above the junction must also be unshaded (no clear path left or right) and hence, we have found the shaded circle in the '1'. After making some more deductions we get to the following situation.

Step 7

The final step is seeing that the path from the shaded circle in the '1' must go left (otherwise we force a closed loop). From here some straightforward deductions lead us to the following finished picture.