# List of Skyscrapers variants

This page lists Skyscrapers variant that has appeared on the WPCs. Roland Voigt's website^{[1]} has a gigantic list of Skyscrapers variations. The variants are listed loosely in chronological order of their invention.

### Skyscraper Sums[edit]

Invented by Tim Peeters (Netherlands) in 1999.^{[2]}

Place a number from 1 to N into each cell so that each number appears exactly once in each row and column. Each number represents a skyscraper of its respective height. The numbers outside the grid indicate the sum of the heights of the skyscrapers that can be seen in the respective row or column from the respective direction.

(Example from PGP IB)

#### Rule variation[edit]

- Puzzles that use multiplication instead of addition are called
**Product Skyscrapers**. Very natural extension, though the first puzzle found is from WPC 2006/Part 6. It is likely that the puzzle predates this.

### Mixed Information[edit]

Hybrid with Easy as puzzle. The earliest example found is a puzzle in the 8th 24hPC^{[3]} written by Bernhard Seckinger (Germany).

Place a number from 1 to N into each cell so that each number appears exactly once in each row and column. Each number represents a skyscraper of its respective height. The numbers outside the grid indicate either how many skyscrapers can be seen in the respective row or column from the respective direction, or the first skyscraper in the respective row or column, or possibly both.

(Example from WPC 2019 IB)

### Skyscraper Cluster[edit]

First appeared on LMD portal^{[4]} in 2010. Invented by Christian Halberstadt (Germany).

Enter numbers into the grid, so that each region of size N×N contains numbers from 1 to N, and within a region. Each number appears exactly once in each row and column. These numbers represent skyscrapers of the corresponding height. Numbers bordering another grid are also clues for the adjacent grid, they indicate how many skyscrapers can be seen in the respective row or column from the respective direction. However, only clues in cells with circles are correct, clues in grey cells are always incorrect. If a corner cell borders two different grids, its clue number must be correct for both adjacent grids if the cell contains a circle, or incorrect for both adjacent grids if the cell is grey. Numbers outside the grid are correct clues for the adjacent grid.

All clue numbers refer only to the adjacent grid, only visible skyscrapers in the adjacent grid are counted.

(Rules and example from WPC 2019 IB)

### Doppelblock Skyscrapers[edit]

*See Doppelblock#Doppelblock_Skyscrapers.*

### Index Skyscrapers[edit]

Invented by Ivan Pastucha (Slovakia) in 2011.

Place a number from 1 to N into each cell so that each number appears exactly once in each row and column. Each number represents a skyscraper of its respective height. The clues inside the grid indicate the number of directions the skyscraper is visible from when looking from the outside of the grid from the corresponding direction. Higher skyscrapers block the view of smaller ones.

(Rules and example from WPC 2016 IB)

### With Mirrors[edit]

The first ruleset (see below) can be seen on the 13th 24-Hour Puzzle Championship (2012). The other two first appeared on WPC 2018/Round 2. All puzzles were written by Jiří Hrdina (Czech Rep.). A set of 3 slightly different rulesets were published under the same name.

Place a number from 1 to N into each cell so that each number appears exactly once in each row and column. Each number represents a skyscraper of its respective height. Additionally, one diagonal mirror is placed in each row and column. Sight is vertically reflected by mirrors.

3 slightly different rules regarding the placement of mirrors were presented. Namely,

- Mirrors are already given.
- Only the positions of the mirrors are given but their orientation is unknown.
- Both the positions and the orientations are unknown.

(Example (first ruleset) from WPC 2018 IB)

### With Glass Towers[edit]

An example from 2012^{[5]} exists, but since this puzzle is a combination puzzle, chances are high that the puzzle predates this.

Place a number from 1 to N into each cell so that each number appears exactly once in each row and column. Each number represents a skyscraper of its respective height. One building in each row and column is invisible. You have to decide where the invisible buildings are.

(Example from WPC 2018 IB, invisible buildings are indicated by grey cells)

### Inside Skyscrapers[edit]

Different variants with this name exist. One is equivalent to Toroidal version of Stroll among Skyscrapers (see below). This version first appeared on 2012 Indian Nationals.^{[6]} Author of the puzzle was Deb Mohanty (India).

Place a number from 1 to N into each cell so that each number appears exactly once in each row and column. Each number represents a skyscraper of its respective height. The clues on the edges between some cells indicate the number of skyscrapers visible in corresponding row or column from that point.

(Example from WPC 2017 IB)

### Skyscrapers^2[edit]

This name was given when Prasanna Seshadri (India) invented the puzzle for WPC 2017. However, independently Christian Halberstadt (Germany) also thought of the same rules and posted the puzzle in the LMD portal under the title *Kleine Hochhausserie: Hoch angesehene Hinweise* ("Small Skyscrapers series: clues that look at height"). ^{[7]}

Place a number from 1 to N into each cell so that each number appears exactly once in each row and column. Each number represents a skyscraper of its respective height. The clues on the edges between some cells indicate the number of skyscrapers visible in the corresponding row or column.

Numbers beside diagonal lines indicate the number of skyscrapers seen considering skyscraper clues in a line in the corresponding direction. It may be part of solving to use the missing skyscraper clues.

(Example from WPC 2017 IB)

### Skyscrapers Ltd[edit]

First appeared on WPC 2016/Round 11. Invented by Michal Forišek (Slovakia).

Place a number from 1 to N into each cell so that each number appears exactly once in each row and column. Each number represents a skyscraper of its respective height.

One building in each row and column is owned by *Skyscrapers Ltd. corporation*. All of those buildings have different heights. A number outside the grid in a circle means that such a building must be visible from that direction. A number outside the grid without a circle means that such a building must not be visible from that direction.

(Example from WPC 2016 IB)

### With GT Hints[edit]

First appeared on WPC 2018/Round 2.

Place a number from 1 to N into each cell so that each number appears exactly once in each row and column. Each number represents a skyscraper of its respective height. In each row/column, the relation between the number of visible building and the height of the first seen building is given.

(Example from WPC 2018 IB)

### Skyscrapers / As Easy As ABC[edit]

First appeared on WPC 2018/Round 2. Hybrid with Easy as Rules.

Place a number from 1 to N into each cell so that each number appears exactly once in each row and column. Each number represents a skyscraper of its respective height. Some cells remain empty. The numbers indicate the sum of the numbers of visible buildings and the first building seen.

(Example from WPC 2018 IB)

### Myopia[edit]

First appeared on WPC 2018/Round 2. Hybrid with Myopia arrows.

Place a number from 1 to N into each cell so that each number appears exactly once in each row and column. Each number represents a skyscraper of its respective height. The arrows in the cells indicate all the directions from which the most number of buildings can be seen.

(Example from WPC 2018 IB)

### Stroll among Skyscrapers[edit]

First appeared on WPC 2018/Round 2.

Place a number from 1 to N into each cell so that each number appears exactly once in each row and column. Each number represents a skyscraper of its respective height. The numbers in the grey cells indicate the number of visible skyscrapers in the direction of the arrow. There are no buildings in the grey cells.

(Example from WPC 2018 IB)

### First Invisible[edit]

First appeared on WPC 2018/Round 2.

Place a number from 1 to N into each cell so that each number appears exactly once in each row and column. Each number represents a skyscraper of its respective height. The numbers outside indicate the first invisible skyscrapers from the given direction.

(Example from WPC 2018 IB)

### With Sum Baskets[edit]

First appeared on WPC 2018/Round 2.

Place a number from 1 to N into each cell so that each number appears exactly once in each row and column. Each number represents a skyscraper of its respective height. The numbers outside the grid indicate how many skyscrapers can be seen in the respective row or column from the respective direction. For each outlined regions, the sum of the heights of the skyscrapers in the region is given. (This is different from Killer rules because of the absence of non-repeating rule in regions)

(Example from WPC 2018 IB)

### Sky Battleships[edit]

*See List of Battleships variants#Sky Battleships.*

### Double Skyscrapers[edit]

First appeared on WPC 2018/Round 10. Written by Jiří Hrdina (Czech Rep.).

Place a number from 1 to N into each cell of the two empty grids, so that each number appears exactly once in each row and column of each grid. Each number represents a skyscraper of its respective height. The numbers outside the third grid indicate the total number of skyscrapers that can be seen in the respective row or column from the respective direction in each of the grids.

#### Rule Variation[edit]

- In one of the puzzles in the same round, there was an additional rule that states skyscraper heights in the corresponding cells add up to 6. (N=5)

## Appearances in the past WPCs[edit]

- WPC 2019/Round 2 (Mixed Information) by Gabi Penn-Karras
- WPC 2019/Round 4 (Doppelblock Skyscrapers) by Roland Voigt
- WPC 2019/Round 5 (Skyscraper Cluster) by Christian Halberstadt
- WPC 2019/Team Playoffs (Doppelblock + Skyscrapers Sums) by Rainer Biegler
- WPC 2018/Round 3 (Mirrors, GT Hints, As Easy As ABC, Myopia, Glass Towers, First Invisible, Sum Baskets, Snail, Coral, Battleships) by Jiří Hrdina
- WPC 2018/Round 10 (Double Skyscrapers) by Jiří Hrdina
- WPC 2017/Round 1 (Skyscraper Sums) by Deb Mohanty
- WPC 2017/Round 4 (Inside Skyscraper) by Deb Mohanty
- WPC 2017/Round 20 (Skyscrapers
^{2}) by Rohan Rao and Prasanna Seshadri - WPC 2016/Round 5 (Index Skyscrapers) by Matej Uher
- WPC 2016/Round 11 (Skyscrapers Ltd) by Michal Forišek and Matej Uher
- WPC 2015/Round 3 (Product Skyscrapers) by Andrey Bogdanov

## References[edit]

- ↑ https://www.hochhausigel.de/index.php
- ↑ http://www.oocities.org/kostunix/oldindex.html
- ↑ http://rejtveny.atw.hu/24HPC2007_booklet_full.pdf
- ↑ https://logic-masters.de/Raetselportal/Raetsel/zeigen.php?chlang=en\&id=0000TE
- ↑ https://logic-masters.de/Raetselportal/Raetsel/zeigen.php?id=00019Y
- ↑ https://logicmastersindia.com/lmitests/?test=IPC2012
- ↑ https://logic-masters.de/Raetselportal/Raetsel/zeigen.php?chlang=de&id=0001TM