Skyscrapers is one of my favourite Japanese logic puzzles because it forces you to think three-dimensionally, and also because Tokyo is full of skyscrapers.
The aim is to fill in each grid below with the numbers from 1 to5 such that every number appears only once in every row and column (like Sudoku). I’ve included an easy, a medium and a hard.
Now for the rules. Imagine each grid is an aerial view of a city block of skyscrapers of varying heights, one within each cell in the grid. The number in a cell indicates the height of the skyscraper in that cell.
A number outside the grid describes how many skyscrapers can be seen along that row or up/down that column from the perspective of that number on the ground. You can only see a skyscraper if smaller skyscrapers are in front of it; you cannot see a skyscraper if a taller one is in front of it, blocking the view.
That’s it! On your marks, get set, solve!
You can copy out the grids by hand, or click here for a printable page. If you are reading this on a mobile device, enlarge the image of each grid, make a screen grab, and write on the grid with an app like Markup, which is contained within Photos on your iPhone.
Scroll below for some tips on how to solve them, but first a historical interlude:
Skyscrapers was invented in 1992 by Masanori Natsuhara. It was first published in the section for reader submissions in Puzzler, a now defunct Japanese puzzle magazine that pioneered many types of grid puzzle. Skyscrapers has since become a Japanese classic, and the grids above were designed by hand by the staff of Toketa, a Japanese puzzle fanzine.
And a commercial interlude: Skyscrapers is one of the twenty six types of Japanese logic puzzle in Puzzle Ninja, my book on the Japanese puzzle scene, which is just out in paperback in the UK, and newly released in the US. (It includes different examples to the one on this page). I recommend it for lazy summer days!
And now for the tips. I’ll help you solve the example below. Puzzle A is 4×4 so the numbers required are the numbers from 1 to 4.
B) The 4 to the left of the second row means we can see 4 skyscrapers, which means that they have to be in the order 1-2-3-4. The 1 at the bottom means we can see only 1 skyscraper, so the first one in view must be a 4, the tallest possible skyscraper.
C) Since all numbers appear once in every row and column, we know that there is a 4 in the first column. It cannot be in the bottom cell, since that would mean two 4s are in the bottom row, which is forbidden. Nor can it be in the third cell down, since that would mean you can only see a single skyscraper along the third row from the left, but the clue says you can see 2. So the 4 must be in the top cell, and this fixes the 4 in the third column.
D) There are only three possibilities for the remaining cells in the top row, such that you can see three skyscrapers from the 3 on the right of the grid: 2-3-1, 1-3-2 and 3-1-2. We can eliminate the first two, since they double up digits in either the second or third column. The top line must be 4-3-1-2
E) The first column must be 4-1-2-3, since if the 2 and the 3 were the other way around you would be able to see three skyscrapers from the bottom, not 2 as is marked.
F) The grid now fills in like a Sudoku, based on rule that every number appears only once in each row or column.
Enjoy! These puzzles are extremely satisfying and addictive!
I’ll post the answer grids at 5pm UK time this afternoon.
Thanks to Toketa for supplying the puzzles.
The cheapest place to pick up Puzzle Ninja is The Guardian Bookshop where it costs a mere £5.99, which works out as about 3p a puzzle, Bargain! (RRP £8.99. Free UK p&p over £10, online orders only.) If you are in the US, you can get a copy here.