# Doppelblock

## Rules

Place either a block or a number from 1 to X (integers only) into each cell so that each number appears exactly once in each row and each column. (X is two fewer than the number of cells in each row.) Each row and each column will therefore have exactly two cells with blocks in them. The numbers outside the grid indicate the sum of the numbers between the two blocks in that row or column. Some cells may already be filled in for you.

(Rules and example from PGP IB)

## History of the puzzle

First appeared in online qualifier for Japan Puzzle Championship 2006. Author is unknown. Originally named ビトウィーン・サム ("Between Sum").

Also independently invented by Naoki Inaba (Japan) in 2008 under the name ハサムサム ("Sandwiched Sum"). Original set of puzzles by Naoki can be seen here

## Variants

Unless noted, the first rule of Doppelblock (placing 1-X and two black cells per row/column) applies.

### Serrated Doppelblock

First appeared in WPC 2019/Round 2. The puzzle was written by Gabi Penn-Karras (Germany).

Enter numbers from 1 to X into the grid and shade the remaining cells, so that each full row and column (consisting of X+2 cells) contains each number exactly once as well as two shaded cells. (X is two fewer than the number of cells in each row.)

The clue numbers next to these rows or columns indicate the sum of the numbers between the two shaded cells in the respective row or column; a sum of 0 means that the shaded cells are adjacent. Incomplete rows (less than X+2 cells) can contain any number of shaded cells; numbers cannot repeat in such a row or column. The clue numbers next to these rows or columns, marked with arrows, indicate the sum of all numbers in the respective row or column.

(Rules (modified) and example (X=3) from WPC 2019 IB)

### Doppelblock Skyscrapers

Hybrid with Skyscrapers puzzle. Invented by Roland Voigt (Germany) in 2014. First appeared in 4th LMD Skyscrapers contest.

The numbers outside the grid indicate how many skyscrapers can be seen from the first shaded cell in the direction and up to the second shaded cell; skyscrapers not between the shaded cells are ignored. A number 0 means the shaded cells are adjacent.

(Rules from WPC 2019 IB, example from 24hPC 2016 IB)

### Doppelblock + Skyscraper Sums

Invented by Rainer Biegler (Germany) in 2014. First appeared on LMD portal.

The numbers outside the grid are both Doppelblock and Skyscraper Sums clues: Each clue number indicates the sum of the numbers between the two shaded cells in the respective row or column. Each clue number indicates the sum of the heights of all visible skyscrapers in the respective row or column, ignoring the shaded cells. Each clue must be valid in both respects.

(Rules and example from WPC 2019 IB)

### Double Doppelblock

First appeared on WPC 2018/Round 10, "Double Trouble". The puzzle was written by Jiří Hrdina (Czech Rep.).

In the given grid, half of the clues belong to one Doppelblock puzzle and the other half belong to the other. Furthermore, all the cells where both puzzles does not contain numbers are marked in the given grid. Reconstruct two puzzles and solve them.

(Example from WPC 2018 IB)