# Plow-Pair

Futoshiki is fun. It’s a number-puzzle where you use logic to re-create a 5×5 square in which every row and column contains the numbers 1 to 5. At first, most or all of the numbers are missing. You work out what those missing numbers are by using the inequality signs scattered over the futoshiki. Here’s an example:

There are no numbers at all in the futoshiki, so where do you start? Well, first let’s establish some vocabulary for discussing futoshiki. If we label squares by row and column, you can say that square (4,5), just above the lower righthand corner, dominates square (4,4), because (4,5) is on the dominant side of the inequality sign between the two squares (futōshiki, 不等式, means “inequality” in Japanese). Whatever individual number is in (4,5) must be greater than whatever individual number is in (4,4).

Conversely, you can say that (4,4) is dominated by (4,5). But that’s not the end of it: (4,4) is dominated by (4,5) but dominates (3,4), which in its turn dominates (2,4). In other words, there’s a chain of dominations. In this case, it’s a 4-chain, that is, it’s four squares long: (4,5) > (4,4) > (3,4) > (2,4), where (4,5) is the start-square and (2,4) is the end-square. Now, because 5 is the highest number in a 5×5 futoshiki, it can’t be in any square dominated by another square. And because 1 is always the lowest number in a futoshiki, it can’t be in any square that dominates another square. By extending that logic, you’ll see that 4 can’t be in the end-square of a 3-chain, (a,b) > (c,d) > (e,f), and 2 can’t be in the start-square of a 3-chain. Nor can 3 be in the start-square or end-square of a 4-chain.

Using all that logic, you can start excluding numbers from certain squares and working out sets of possible numbers in each square, like this:

[whoops: square contains errors that need to be corrected!]

Now look at column 1 and at row 4:

In column 1, the number 5 appears only once among the possibles, in (1,1); in row 4, the number 1 appears only once among the possibles, in (4,1). And if a number appears in only one square of a row or column, you know that it must be the number filling that particular square. So 5 must be the number filling (1,1) and 1 must be the number filling (4,1). And once a square is filled by a particular number, you can remove it from the sets of possibles filling the other squares of the row and column. I call this sweeping the row and column. Voilà:

Now that the 5 in (1,1) and the 1 in (4,1) have swept all other occurrences of 5 and 1 from the sets of possibles in column 1 and row 4, you can apply the only-once rule again. 2 appears only once in row 4 and 5 appears only once in column 4:

So you’ve got two more filled squares:

Now you can apply a more complex piece of logic. Look at the sets of possibles in row 3 and you’ll see that the set {2,3} occurs twice, in square (3,1) and square (3,4):

What does this double-occurrence of {2,3} mean? It means that if 2 is in fact the number filling (3,1), then 3 must be the number filling (3,4). And vice versa. Therefore 2 and 3 can occur only in those two squares and the two numbers can be excluded or swept from the sets of possibles filling the other squares in that row. You could call {2,3} a plow-pair or plow-pare, because it’s a pair that pares 2 and 3 from the other squares. So we have a pair-rule: if the same pair of possibles, {a,b}, appears in two squares in a row or column, then both a and b can be swept from the three other squares in that row or column. Using {2,3}, let’s apply the pair-rule to the futoshiki and run the plow-pare over row 3:

Now the pair-rule applies again, because {4,5} occurs twice in column 5:

And once the plow-pare has swept 4 and 5 from the other three squares in column 5, you’ll see that 3 is the only number left in square (1,5). Therefore 3 must fill (1,5):

Now 3 can be swept from the rest of row 1 and column 5:

And the pair-rule applies again, because {1,2} occurs twice in row 2:

Once 2 is swept from {2,3,4} in square (2,1) to leave {3,4}, 3 must be excluded from square (2,2), because (2,2) dominates (2,1) and 3 can’t be greater than itself. And once 3 is excluded from (2,2), it occurs only once in column 2:

Therefore 3 must fill (5,2), which dominates (5,1) and its set of possibles {2,3,4}. Because 3 can’t be greater than 4 or itself, 2 is the only possible filler for (5,1) and only 3 is left when 2 is swept from (3,1):

And here are the remaining steps in completing the futoshiki:

The complete futoshiki

Animation of the steps required to complete the futoshiki

Afterword

The pair-rule can be extended to a triplet-rule and quadruplet-rule:

• If three numbers {a,b,c} can occur in only three squares of a row or column, then a, b and c can be swept from the two remaining squares of the row or column.
• If four numbers {a,b,c,d} can occur in only four squares of a row or column, then a, b, c and d can be swept from the one remaining square of the row or column (therefore the number e must fill that remaining square).

But you won’t be able to apply the triplet-rule and quadruplet-rule as often as the pair-rule. Note also that the triplet-rule doesn’t work when {a,b,c} can occur in only two squares of a row or column. An n-rule applies only when the same n numbers of a set occur in n squares of a row or column. And n must be less than 5.

Post-Performative Post-Scriptum

Domination. Exclusion. Inequality. — an earlier look at futoshiki

# Domination. Exclusion. Inequality.

Sudoku has conquered the world, but I think futoshiki is more fun — more concentrated, more compact and quicker. When complete, a 5×5 futoshiki will be a Roman square in which every row and column contains the numbers 1 to 5, but no number is repeated in any row or column. You have to work out the missing numbers using logic and the “inequality” signs that show whether one square contains a number more than or less than a number in a neighboring square — futōshiki, 不等式, means “inequality” in Japanese (fu- is the negative prefix). Here’s an example of a futoshiki puzzle:

Futoshiki puzzle, with 4 in square (3,2) and 3 in square (1,1)

If you identify the squares by row and column, 4 is in (3,2) and 3 is in (1,1). And you can say, for example, that the empty square (3,4) dominates the empty square (3,3) or that (3,3) is dominated by (3,4). I’ll describe one route (not the best or most efficient) to completing the puzzle. Let’s start by considering the general rules that 1 cannot appear in any square that dominates another square and that 5 cannot appear in any square dominated by another square.

If you extend that logic, you’ll see that 4 cannot appear in any square that is at the end of what you might call a chain of dominations, where one square dominates a second square that in turn dominates a third square. Therefore, in the puzzle above, 4 cannot appear in squares (2,1) and (4,1) of column 1. And it can’t appear in square (3,1), because that would mean two 4s in the same row. This leaves one place for 4 to appear: square (5,1). And if 4 is there, 5 has to be in square (3,1):

Now look at row 3. Two of the remaining three empty squares are dominators: (3,4) dominates (3,2) and (3,5) dominates (2,5). 1 cannot appear in a dominating square, so 1 has to be in the dominated square (3,3):

The next step I’ll take is a bit more complicated. In row 2, the number 4 cannot be in (2,1) and (2,4). It can’t be in (2,1) because that would mean 4 was greater than itself. And it can’t be in (2,4), because (2,4) is dominated by (3,4) and (3,4) can’t contain 5, the only number that dominates 4. Therefore 4 must be in either (2,3) or (2,4). But so must 5. Therefore (2,3) contains either 4 or 5 and (2,4) contains either 4 or 5. That means that the numbers [1,2,3] must be in the other three squares of row 2. Now, 3 can’t be in square (2,1), because the chain of dominations is too long. And 3 can’t be in (2,5), because (2,5) is dominated by (3,5), which contains either 2 or 3. Therefore 3 must be in (2,2):

Now consider column 2. Square (4,2) cannot contain 1 or 5, because it’s both dominated and dominating. And if it can’t contain 1 or 5, there’s only one number it can contain: 2. And it immediately follows that (4,1) must contain 1, the only number less than 2. And if four squares of column 1 now contain the numbers [1,3,4,5], the remaining empty square (2,1) must contain 2:

Now consider row 2. Squares (2,3) and (2,4) contain either 4 or 5, therefore (2,5) must contain 1:

Now consider row 5. The number 1 is logically excluded from three squares: from (5,3), because there’s a 1 in (3,3); from (5,4), because (5,4) dominates (5,5); and from (5,5), because there’s a 1 in (2,5). Therefore 1 must be in (5,2). And if 1 is in (5,2), the number 5 must be in (1,2):

Now 1 pops up in row 1 because it can only be in (1,4):

And 5 pops up in row 4 because it can only be in (4,5):

Once 5 is in (4,5), the number 4 must be in (4,4) and the number 3 in (4,3):

The 4 of (4,4) immediately collapses the ambiguity of (2,4), which must contain 5. Therefore (2,3) contains 4:

Next, 5 pops up in (5,3):

And 3 must be in (5,4), dominating 2 in (5,5):

With 3 in (5,4) and 2 in (5,5), the ambiguity of (3,4) and (3,5) collapses:

And the square is completed like this:

Here’s an animated version of the steps to completion:

Futoshiki puzzle animated

# Fourtoshiki

I hadn’t realized that sudokus could be witty until earlier this year, when I did one that literally made me laugh, because the solutions were so clever and quirky. Foolishly, I neglected to make a note of the sudoku so I could reproduce it. But I haven’t made that mistake with this futoshiki:

Using more-than and less-than signs to deduce values, fill each line and column with the numbers 1 to 5 so that no number occurs twice in the same row or column

It’s not witty like that lost sudoku, but I think futoshikis are even more beautiful and enjoyable than sudokus, because they’re even more elemental. They’re also rooted in the magic of binary, thanks to the more-than / less-than clues. And when there’s only one number on the original grid, completing them feels like growing a flower from a seed.