Inequality Number Puzzles: How to Solve Greater-Than Grids

To solve an inequality number grid, use every sign to strike out the digits a cell cannot hold, then follow chains of signs, because a long chain pins down its end cells almost at once. Combine those limits with the rule that each digit appears once in every row and every column, and the grid fills itself in a steady order.

This guide explains how to read the signs, how chains and valleys work, a mirror trick for checking your thinking, a full worked example and a practice puzzle with its answer.

What is an inequality number grid?

An inequality number grid is a square grid, such as four by four or five by five, that you fill with the digits from 1 up to the size of the grid. As in any Latin square, each digit appears exactly once in every row and exactly once in every column.

The twist is a set of small signs placed between some neighbouring cells. A sign between two cells in a row is a greater-than or less-than sign. A sign between two cells in a column is the same sign turned on its side. Some puzzles also give a few digits to start with, and some give none at all, so that the signs alone decide everything.

How do you read the signs?

There is one rule to remember: the point of a sign always aims at the smaller cell, and the open side faces the larger one. In this guide, a sign pointing up between two rows means the upper cell is smaller, and a sign pointing down means the upper cell is larger.

That first pass is easy and worth doing for every sign before anything else. Any cell that is smaller than a neighbour can never be the largest digit, and any cell that is larger can never be 1.

Why are chains of signs so powerful?

A chain is a run of signs that all point the same way, such as three cells where the first is smaller than the second and the second is smaller than the third. Chains can bend round corners, because each sign only links two cells. Every step of a chain goes strictly up, so all of its cells hold different digits, even when the chain turns a corner, and each step adds at least one.

The general rule is simple: a cell with k cells below it in a chain must be at least k plus 1, and a cell with k cells above it can be at most the grid size minus k. Count the cells on each side, and the limits follow at once.

What does a valley or a peak tell you?

A valley is a cell that is smaller than two neighbours in the same row or column. Since those two neighbours hold two different digits, both larger than the valley, the valley cannot be either of the two largest digits. In a four by four grid, a valley is always 1 or 2.

A peak is the reverse: a cell larger than two neighbours in one line. It can never be 1 or 2 in a four by four grid. Valleys and peaks become even stronger once a few digits in that line are placed, because then the leftover options can allow only one choice.

How do the signs act like forbidden entries?

Each sign shrinks the list of digits a cell may hold. Mathematicians look at a different and only loosely related question about Latin squares with forbidden entries.

A paper published on 12 May 2006 calls an n by n array avoidable when some Latin square of the same size differs from it in every cell, so that in every filled cell of the array the square holds a different symbol ^[1]. That paper traces the question of which arrays can be avoided back to 1989 ^[1].

The paper also restates a known result for partial Latin squares, meaning arrays in which each cell is empty or holds one symbol and no symbol repeats in a row or column ^[1]. Every such array of order 4 or more is avoidable ^[1]. The paper adds that this fails for order 3, where some partial Latin squares cannot be avoided ^[1].

The link to sign puzzles is our own loose comparison, not something that paper discusses. A sign forbids an order between two cells rather than one symbol in one cell. Still, once you have struck out the impossible extremes, each cell carries a short list of banned digits, a little like those arrays. You then work the other way round from avoidance, striking out digits until only one Latin square is left.

Can you rename the digits in a sign puzzle?

In a plain Latin square you can rename the symbols freely and nothing important changes. Mathematicians group squares that turn into each other by reordering rows, reordering columns, renaming symbols or flipping the grid across its main diagonal into one transposition class ^[2]. A paper published on 1 March 2013 uses this grouping and lists a representative for each transposition class of order 4, and there are only two of them ^[2].

Signs break most of that freedom, because renaming digits can ruin the order between them. One relabelling always survives: swap every digit d for the grid size plus 1 minus d, so that in a four by four grid 1 becomes 4, 2 becomes 3 and so on, and turn every sign round. The new puzzle has exactly one solution whenever the old one did.

This mirror trick is a handy check on your reasoning. If a deduction about small digits feels shaky, think through the mirror version about large digits instead. The two arguments are the same, so one of them may look clearer.

How does the method solve a four by four puzzle?

Here is an original puzzle with no given digits at all, only seven signs. Fill the grid with 1 to 4, once in every row and every column. Columns are lettered A to D and rows are numbered 1 to 4.

In words, the signs say: B1 > C1, C1 > D1, D1 > D2, A3 > A2, C2 > C3, C4 > C3 and D4 > D3. We checked by computer, over every four by four Latin square, that this puzzle has exactly one solution.

Step 1: follow the long chain. Start at D2 and climb: D2 is smaller than D1, D1 is smaller than C1, and C1 is smaller than B1. That is four cells linked by three signs that all point the same way, so the four digits climb strictly. Four climbing digits in a four by four grid must be 1, 2, 3 and 4. So D2 is 1, D1 is 2, C1 is 3 and B1 is 4.

Row 1 then needs a 1, so A1 is 1.

Step 2: column D. Column D holds 2 and 1, so D3 and D4 share 3 and 4. The sign says D3 is the smaller, so D3 is 3 and D4 is 4.

Step 3: the valley in column C. Column C holds 3 at the top, so C2, C3 and C4 share 1, 2 and 4. C3 is smaller than both C2 and C4, which makes it a valley, so it must be the smallest of the three: C3 is 1.

Step 4: row 3. Row 3 now holds 3 and 1, so A3 and B3 share 2 and 4. Column B already holds 4 in B1, so B3 is 2 and A3 is 4.

Step 5: column B. Column B holds 4 and 2, so B2 and B4 share 1 and 3. Row 2 already has a 1 in D2, so B2 is 3 and B4 is 1.

Step 6: finish rows 2 and 4. Row 2 needs 2 and 4 in A2 and C2. Column A already holds 1 and 4, so A2 is 2 and C2 is 4. Row 4 then needs 3 and 2: column A takes 3 in A4, and C4 gets 2.

Check every sign. 4 > 3 > 2 along row 1, 2 > 1 down column D, 2 < 4 in column A, 4 > 1 < 2 down column C and 3 < 4 in column D. Every row and column holds 1 to 4 once. All seven signs agree.

What are the most common mistakes?

• Reading a sign backwards. Say it aloud once: the point aims at the smaller cell. Take extra care with the turned signs between rows.
• Stopping at the ends of a chain. The middle cells of a chain are limited too, so write their options down as well.
• Forgetting the row and column rule. Use the signs together with the usual Latin square scans of every row and column.
• Skipping the first pass. Striking out the impossible extremes for every sign takes a minute and can reveal a single option straight away.

What should you do when you get stuck?

• Look for the longest chain you have not used yet and write the options for each of its cells.
• Hunt for valleys and peaks, especially in lines that already hold a digit or two.
• Pick a digit, say the largest, and ask where it can still go in each row. A cell that is smaller than a neighbour can never hold it.
• Try the mirror view: think about the largest digits instead of the smallest, or the reverse.

Can you solve this practice puzzle?

Fill the grid with 1 to 4, once in every row and every column. There are six signs and no givens, and we checked by computer that it has exactly one solution.

In words: A1 > A2, B2 > B3, D2 > D3, B3 > A3, A4 > A3 and B4 > A4.

Hints. A3 starts two chains of three, one through B3 to B2 and one through A4 to B4. So B2 and B4 must both be 3 or 4, which uses up 3 and 4 in column B. That leaves B3 as 1 or 2, and since A3 is smaller still, A3 is 1 and B3 is 2. Row 3 then needs 3 and 4 in C3 and D3, and the sign under D2 settles which is which.

Answer:

Ready for a different kind of number reasoning? This riddle asks for a four-digit number whose digits reach a set total and are tied to one another by a few short rules, and it rewards the same careful, one-step-at-a-time thinking.