# Under the Surface

This is part 13 of the puzzle series Around the World in Many Days. Each part is solvable on its own.

Deаr Puzzling,

The first grid is a kakuro puzzle. Fill the grid with the numbers 1–9 so that each run of consecutive numbers (horizontally or vertically) sums up to the number indicated outside the grid without using any digit more than once. Each digit has a background colour (white, grey or red) and each instance of the same digit in the grid has the same colour. In the finished grid, the colours determine what colour to use for each of the digits 1-9 in the second grid.

The second grid forms a Tentai Show puzzle. Along cell borders, divide the grid into rotationally symmetrical “rooms”. Each room must contain exactly one circle, located in the exact middle of the room. Colour every room with the colour indicated by the number inside the circle.

Today I have squeezed into my wetsuit to explore the underwater world of a lovely small island. Can you guess where I am?

Gladys will return in Coasting Along the Coast.

The kakuro can be solved this way:

The 16's in the top left cannot be 8 + 8, so both have to be 7 + 9. To satisfy the 22 clue the right cell must then be a 6, and to satisfy the 29 clue the remaining cells must add up to 13. We've used 7 and 9 so we cannot use 7 + 6 or 9 + 4 to make 13. Hence we must use 5 + 8. To satisfy the vertical 21 clue, the cell below the 6 we just placed must be 8, or else it would be a 5 which means the remaining four cells add up to 7 without repeating digits which is impossible. The 5 we placed makes the cell to its right an 8, and that 8 makes the cells below it a 1 and 2 in some order to satisfy the 11 clue.

Then, if the two cells nearest the 30 clue sum to less than 6, the remaining three cells must add up to more than 24 without repeats, which is impossible since 24 is the most you can make with three different digits (9 + 8 + 7). Hence the two cells nearest the 30 clue sum to exactly 6, meaning they are 4 and 2. This makes the three cells exactly 9, 8, and 7 in some order. The cell below the 2 we placed is thus a 1, and the cells below the 4 cannot be 4 anymore.

The cell nearest the 13 clue must be 7, or else the three cells below it add up to less than 6 without repeats, again impossible (6 is the least you can make with 1 + 2 + 3). Therefore the three cells sum to exactly 6, and hence are 1, 2, and 3 in some order. Now, the horizontal 21 clue has three cells which sum to 6 (there's a 1 placed, and then there's a 2 and 3 in some order). Therefore the remaining cells add up to 15, since this is the minimum sum you can make without 1, 2, or 3. Equivalently, the remaining cells are 4, 5, and 6 in some order.

And now I will decide to use the constraint that each instance of the same digit has the same color. I could have used this a few steps earlier, but I just wanted to use the kakuro constraints first as much as is feasible for me. Anyway, this fills up the grid significantly:

So now we know that 1, 6, and 9 are red; 2, 4, and 7 are gray; and 3, 5, and 8 are white. Moving on, this implies that the second-farthest cell from the 35 clue has to be 8 as it is white and sees 3 and 5. Likewise, the cell below it is a 6 as it is red and sees 1 and 9. We have 9 + 5 + 8 + 6 = 28, so the top two cells sum to 7. They are white and gray, so it is either 2 + 5 or 3 + 4, but the 5 has already been used in the column so it has to be 3 + 4. This resolves the top right 14 clue; both remaining cells are gray and sum to 6, so they must be 2 + 4 and we know the order due to the 4 we just placed. We can also disambiguate the bottom right 14 clue, and then the bottom left cells in red, and so the kakuro is complete.

We now move on to the Tentai Show puzzle:

I hate to say this, but I will leave some deductions to the reader as an exercise. The deductions are not too hard, though, and I will still give an outline of my solving process which is simple. First, from the constraint that we need to preserve rotational symmetry alone, we can mark many edges.

Next, we look at cells which can each be claimed by only one clue: A can only be claimed by 1 above it, B by the 2 to its right, C by the 1 to its left, D by the 9 two cells to the left and two cells below, and E by the 5 touching its top left corner. In these cases, the reasons why each of them cannot be claimed by any other clue is one of the following three: either 1) the mirror image of the cell cannot be claimed by a clue without disobeying rotational symmetry, 2) the mirror image of the cell is already occupied by another clue, or 3) connecting both the cell and its mirror image to the center of the clue cannot be done without breaking the rules. From here on out, we can further infer which cells have to belong to which clues, and eventually we progress far enough to finish the puzzle:

Now, if we apply the coloring for each number, this graphic is shown:

which slightly resembles the flag of Malta... but there also seems to be some sort of lettering overlaid on it. This seems to be a big lowercase cursive G, followed by an o, then a z and o (which I thought was '20' at first). This spells out Gozo, and indeed as can be seen in the link above Gozo is an island that's part of the Republic of Malta. This also checks out with Gladys's account of her travel. And so the puzzle is solved!