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Skyscrapers ビルディング M A S A N O R I N AT S U H A R A , 1 9 9 2

The challenge: Fill the grid with numbers, so that every number appears only once in every row and column. The numbers used range from 1 up to the length of each row or column.

The rules: Imagine the grid is the aerial view of a city block of sky-scrapers of varying heights, one within each cell in the grid. Each skyscraper is to be represented by a number indicating its height. A number outside the grid describes how many skyscrapers can be seen along that row or up/down that column from the perspec-tive of that number on the ground. You can only see a skyscraper if smaller skyscrapers are in front of it; you cannot see one if a taller skyscraper is in front of it, blocking the view.

How to solve it:A. The starting grid.B. The obvious places to start are either where there is a 4,

meaning we can see all four skyscrapers, or where there is a 1, meaning we can see only one. If we can see every skyscraper they must be arranged in ascending order, so the view from the 4 on the second row must go 1-2-3-4. Likewise, the only way to see a single skyscraper is if the highest comes first, so we can put a 4 above the 1 on the bottom row, since the view from that 1 goes up the second column.

C. With two 4s already in the grid, there are only two positions left for the 4 on the top row: in the first or the third column. We can eliminate the latter since a 4 here would mean it is impossible to see three skyscrapers looking along the top row from the 3. So the 4 is in the first column and the final 4 in the grid must be row three, column three.

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D. In order to see three skyscrapers from 3, one of them must be hidden. The order for the row must be either 4-2-3-1, 4-1-3-2 or 4-3-1-2. We can eliminate the first one since this doubles up 2s in the second column, and we can eliminate the second since this doubles up 3s on the third column. So the order is 4-3-1-2.

E. In order to see only two skyscrapers up the first column, a 3 must be in the bottom cell. The rest of the grid now fills itself based on the rule that every number appears once on each row and column.

F. The completed grid.

D E F

S K YS C R A P E R S

A B C

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D. In order to see three skyscrapers from 3, one of them must be hidden. The order for the row must be either 4-2-3-1, 4-1-3-2 or 4-3-1-2. We can eliminate the first one since this doubles up 2s in the second column, and we can eliminate the second since this doubles up 3s on the third column. So the order is 4-3-1-2.

E. In order to see only two skyscrapers up the first column, a 3 must be in the bottom cell. The rest of the grid now fills itself based on the rule that every number appears once on each row and column.

F. The completed grid.

D E F

S K YS C R A P E R S

A B C

Puzzle example: Solution:

1/ 2/ 3/

1 8 8

T H E YO U N G S A M U R A I

1 / 2 /

3 / 4 /

Y U K I K AWA B E Y U K I K AWA B E

E D E N TA K E YA S A I K AC H I

5 / S H I N I C H I AO K I

1 8 8

T H E YO U N G S A M U R A I

1 / 2 /

3 / 4 /

Y U K I K AWA B E Y U K I K AWA B E

E D E N TA K E YA S A I K AC H I

5 / S H I N I C H I AO K I

1 8 8

T H E YO U N G S A M U R A I

1 / 2 /

3 / 4 /

Y U K I K AWA B E Y U K I K AWA B E

E D E N TA K E YA S A I K AC H I

5 / S H I N I C H I AO K I

More Skyscrapers and other Japanese puzzles can be found in Alex Bellos’s latest book, Puzzle Ninja.