Hi guzzlers

Today’s puzzle comes from Istanbul. Tapa – it stands for Turkish Art Paint – is a fantastic logic grid puzzle, invented by Turkish puzzle master Serkan Yürekli. The puzzle is now a classic in the world of competitive puzzling.

(Yes, competitive puzzling! This week, Britain’s best will be flocking to a hotel in Croydon for the UK Puzzle and Sudoku Tournament 2018.)

I have four Tapa examples of increasing difficulty. Since you are required to use a pencil to fill them in, you will want to print them out, and in that case click here for a printable version. Here’s an example, together with its solution:

The rules are very simple, if slightly cumbersome to explain for the first time. The challenge is to shade in certain cells to create a path of shaded cells such that:

- A number in a cell describes the number of shaded cells among the immediate neighbours of that cell, meaning the ones that are horizontally, vertically and diagonally adjacent to it. The shaded cells must be consecutive, so if the number in the cell is 3, then the three shaded cells must be connected
- If there is more than one number in the cell, then there must be a white cell between the groups of consecutively shaded cells.
- Cells with a number cannot be shaded.
- All the shaded cells must be connected horizontally or vertically in the final grid.
- The shaded cells cannot form a 2×2 square anywhere in the grid.
- There is a unique solution.

I’ll run through the example above so you get the hang of it. The first thing to understand is that each number is **only** referring to the cells that are its neighbours, which are those ones it touches horizontally, vertically and diagonally. So, as illustrated below right, the 3-cell in the corner is only describing the three cells that are its neighbours, the 4-cell on the side is only describing the five cells that are its neighbours, and the 7-cell is only describing the eight cells that are its neighbours. And so on.

Let’s start with the two corners. The 3-cell in the top left corner describes the number of its neighbours that must be shaded. Since it has only three neighbours, we must shade in all of them. The cell with (1,1) in the bottom right means that among that cell’s neighbours, two single cells must be shaded, and there must be a blank space between them. There is only one way to do this, illustrated below left. Always mark a small cross on a cell that has to remain blank.

Next look at the cell with (1, 1, 1) on the top row. It must have three single shaded cells, with a space between each of them, and there is only one way to do this. We can also fill in the neighbours of the 7-cell. We know it must have 7 neighbours shaded. It has 8 neighbours, but since one of them has a cross on it, only 7 are available, and we can shade them all in.

Look at the cell marked A above. If it is left blank, the shaded cell to the left of it will be disconnected (horizontally and vertically) from all other cells. One of the rules says that in the final grid all shaded cells must be connected horizontally or vertically, so we can deduce that A must be shaded. Likewise B must be shaded since if B was blank it would leave the shaded cell above it disconnected from the other shaded cells. The same logic means we can also shade E and F. Now to the 4-cell on the left side. It has 5 neighbours. We must shade in four of them. Consider this statement the other way around: only one of the neighbours must be left blank. The blank one is either C or D, because if the blank one was any other neighbour then the shaded cells would be broken, and the rule states that the shaded cells must be connected. So we can shade in the 3 neighbours that are neither C nor D.

The grid is filling up. In the grid above left, we deduce that G and H must be shaded, since this is the only way to make sure that the shaded cells determined by 3-cell on the right column will connect to the other shaded squares. Likewise, the only possible positions of the remaining shaded neighbours of the (1, 5) cell are I an J. Finally, K must be left blank since if we shaded it there would be a 2×2 cell of shaded squares, which is forbidden. The remaining squares now fill in themselves in order to keep the shaded path in the final grid connected.

Serkan Yürekli invented the puzzle in 2007 as a take on the well-known Japanese grid puzzle O’Ekaki, in which numbers are placed outside a grid and the solver uses these numbers to shade cells in the grid and reveal an image. (O’Ekaki is also known as paint by numbers, griddler, picross, and nonogram). Serkan wanted to see if he could design a logic painting puzzle where the numbers are *inside *the grid. The puzzle became a hit among the puzzlerati because its rules are simple and it can be prepared in many different sizes and at many different levels.

Serkan is a prolific puzzle inventor, and Tapa is his most famous original puzzle type. He makes a living designing puzzles, as well as organising competitions across Turkey, giving seminars, training teachers and giving courses to gifted kids. You can find more of his Tapa puzzles at the website The Art of Puzzles.

I will be posting the answers on this blog at 5pm this afternoon.

*I set a puzzle here every two weeks on a Monday.** **I’m always on the look-out for great puzzles. If you would like to suggest one, email me.*

*I’m the author of several books of popular maths, including the puzzle books Can You Solve My Problems? and Puzzle Ninja.*