# PSE Advent Calendar 2023 (Day 20): Candles on the tree

This puzzle is part of the Puzzling Stack Exchange Advent Calendar 2023. The accepted answer to this question will be awarded a bounty worth 50 reputation.

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Rules of Akari / Light Up:

• Fill the grid by placing lamps in unshaded cells.
• A number in a cell indicates how many lamps are adjacent to that cell, horizontally or vertically.
• A lamp illuminates every cell it sees horizontally or vertically. Lamps cannot see through shaded cells.
• In the filled grid, every unshaded cell must be illuminated by at least one lamp, and no lamp must be illuminated by another lamp.

Solve on Penpa+ (no answer check)

Starting off:

In the furthest left column, tucked behind the 2, one or the other of the two cells must have a bulb to light the cell left of the 2. If we put a bulb in the cell southwest of the 2, it forces bulbs east of this 2, as well as west of the 3, which combined block two of the three possible bulb locations of the 2 clue in the sixth row from top [h/t to msh210 and justhalf for pointing out a previous error in this presentation]. So we can place a bulb west of the 2 at lower left. We get similar logic near the top, where one of the two cells east of the 2 must have a bulb to light the cell north of the 1. If it is the cell northwest of the 1, the cell south of the 1 must also have a bulb, which forces two bulbs adjacent to the 1 on the 2-1 diagonal clue pair in the bottom right. So the cell north of the top 1 must be a bulb. Looking at the two in the middle left, neither its northeast or southeast cells can be a bulb, since they would illuminate two of three possible locations for bulbs next to the 2. The grid thus far:

Now let's get serious:

Look again at the topmost 2. Its two bulbs cannot be both west and south, since they would block two of the 3-clue's bulb locations. So the cell north of the 2 must be a bulb. However, one of these two cells must be a bulb, meaning that the cells south and east of the 3 must both be bulbs. This solves the 1 clue by the 3. Moreover, the 2 in the 2-1 diagonal pair must have a bulb in the cell to its north, and the 1 in this pair cannot have a bulb east or south. The grid thus far:

Some long runs:

In the column with 2 and 3 clues, we must have a bulb on one in that column, and one bulb on the other clue in the column to the west. This eliminates the possibility of a bulb anywhere else in those columns. In particular, this solves the 2-clue in the middle left (row 6), and also places a bulb two cells south of the 1 in the tree trunk. These force the last bulb location north of the 2-clue at lower left, as it is the only cell that can light itself. The grid thus far:

Hopefully finishing up:

In the lower right corner, the only cell that can light the cell east of the 1 two cells east of the 3 is the cell next to it. Now the only cell that can light the cell south of the 1 in the 2-1 pair is the cell north of the 3-clue, and the rest falls apart.

• "these bulbs combined force the bulbs on the lower right corner 1s to be adjacent" doesn't seem to be correct. But maybe I'm just not seeing it. Dec 20, 2023 at 4:13
• @msh210 I was about to comment on that too, then I saw your comment. Although at least there is an alternative reasoning, being the 2 above will be forced to have north+east, and the 3 at the bottom only has 2 places left. Dec 20, 2023 at 10:43
• @msh210 - Ah, thanks for pointing this out. I had a longer contradiction chain when I did the puzzle, which started the way I presented but ultimately ended the way justhalf suggests. I will go back and edit the explanation. Dec 20, 2023 at 11:09

Once you reach this partial solution:

It looks like there are two solutions:

• In your first one there is an unlit cell two to the left of the bottom-right 1...
– Stiv
Dec 20, 2023 at 11:26
• Look at the square to the right of the 1 that is right of the 3. It is unlit in your first solution.
– fljx
Dec 20, 2023 at 11:27
• Ah, thank you! It's been too long since I've played this. In my mind, if there's a dot then it must be lit some other way. But that doesn't indicate that it has actually been lit by a lamp. Dec 20, 2023 at 20:56