Calculation Cage Puzzles: How to Solve Target-and-Operation Grids

To solve a calculation cage grid, start with the cages that allow only one set of digits, such as single cells, big products and wide differences, then list the possible digit sets for every other cage and cut them down with the rule that each digit appears once in every row and column. Each placed digit shrinks the options in nearby cages, so the grid fills in a chain of small, certain steps.

This guide explains the rules, what each operation tells you, how to list the combinations for a sum, a full worked example and a practice puzzle with its answer.

What is a calculation cage grid?

A calculation cage grid is a square grid, often four by four or six by six, that you fill with the digits from 1 up to the size of the grid. Each digit appears exactly once in every row and exactly once in every column, so the finished grid is a Latin square.

The grid is also divided into outlined groups of cells called cages. Each cage shows a target number and an operation: add, subtract, multiply or divide. The digits in the cage must reach the target with that operation. For subtraction and division, the cage has two cells, and you take the larger digit first.

Digits may repeat inside a cage, but only when the repeated cells share neither a row nor a column. In a straight cage, every digit is different. In an L-shaped cage, the two end cells can match.

What does each operation tell you?

The operation is the strongest clue in a cage. Some targets allow just one pair of digits, and those are the best places to start. Here is what common two-cell cages in a straight line allow in a four by four grid:

Notice the pattern. Large sums, large products and wide differences are rare, so they pin digits down fast. Small differences such as 1 − are the weakest clues and are best left until late.

How do you list the combinations for a sum?

An addition cage asks you to split its target into as many parts as the cage has cells. Mathematicians call such a split a partition: the parts are positive whole numbers, and only which parts are used matters, not their order. For a cage you never need every split of the target, only those with exactly as many parts as the cage has cells.

For a cage, two extra limits apply: no part may be bigger than the grid size, and parts may repeat only in cells that share no line. Take a three-cell cage with target 6 in a four by four grid. Six splits into exactly three parts in three ways: 4 + 1 + 1, 3 + 2 + 1 and 2 + 2 + 2.

In a straight cage, only 3 + 2 + 1 survives, because the other two splits repeat a digit along one line. In an L-shaped cage, 4 + 1 + 1 is also possible, with the two 1s in the end cells. The split 2 + 2 + 2 is never possible, because a connected three-cell cage always has two cells in the same line.

Write these short lists in a corner of the cage. They are your starting candidates.

Which cages should you fill first?

Work from the strongest clue to the weakest. This order is a good default:

1. Single-cell cages. The target is the digit. Write it in at once.
2. One-pair cages. A cage like 12 × or 3 − in a four by four grid has just one pair. You may not know the order yet, but you know which two digits are used, and that removes them from the rest of the line.
3. Order from a neighbour. When one digit of a pair is already in that row or column, the order of the pair is settled.
4. Line totals. In a four by four grid every row and every column adds up to 10. If a cage fills most of a line, the leftover cell is 10 minus the cage total.
5. Weak cages last. By the time you reach a 1 − or a 2 ÷ cage, the row and column rule has usually decided it.

How does the method solve a four by four puzzle?

Here is an original puzzle. Fill the grid with 1 to 4, once in every row and column. Each letter marks a cage, and the table lists the target for each cage. Columns are lettered A to D and rows are numbered 1 to 4.

We checked by computer, with this guide's own short program, that this puzzle has exactly one solution. Have a go first.

Step 1: the single cell. Cage h is just C4, so C4 is 1.

Step 2: cage b. A difference of 3 needs 1 and 4. Column C already has 1 in C4, so C1 is 4 and B1 is 1.

Step 3: cage f. A product of 12 needs 3 and 4. Column C already has 4 in C1, so C3 is 3 and B3 is 4.

Step 4: cage d. A product of 6 needs 2 and 3. Column C already has 3 in C3, so C2 is 2 and B2 is 3. Column C is now complete.

Step 5: finish column B. Column B holds 1, 3 and 4, so B4 is 2.

Step 6: cage g. Two cells in one column adding to 6 must be 2 and 4, since 3 and 3 would repeat. Row 4 already has 2 in B4, so D4 is 4 and D3 is 2.

Step 7: cage e. Row 4 now holds 2, 1 and 4, so A4 is 3. The cage adds to 6, and A4 plus B4 is 5, so A3 is 1.

Step 8: the top corner. Row 1 holds 1 and 4 already, so A1 and D1 share 2 and 3. Cage a needs a ratio of 2, so A1 is 1, 2 or 4, and only 2 fits. So A1 is 2 and D1 is 3. A2 must then be 1 or 4, and column A already has 1 in A3, so A2 is 4. Cage c then needs D2 to differ from 3 by 2, and D2 is 1.

Check every cage. 4 ÷ 2 is 2. 4 − 1 is 3. 3 − 1 is 2. 3 × 2 is 6. 1 + 3 + 2 is 6. 4 × 3 is 12. 2 + 4 is 6. And C4 is 1. Every row and column holds 1 to 4 once.

What do mathematicians say about small blocks of a Latin square?

Square cages of four cells raise a natural question: can the rows of a finished grid be paired up, and the columns too, so that the 2 by 2 blocks where a row pair meets a column pair never repeat a digit? The paired rows or columns need not sit next to each other.

A paper published on 1 March 2013 looked at a version of this question in which the rows and the columns of an even-sized Latin square are paired up, and each row pair meets each column pair in a 2 by 2 block ^[1].

For even orders from 4 to 8, the paper showed that the rows and columns of every Latin square can be paired so that no block repeats a symbol, except for the squares in five transposition classes of order 6, which cannot be paired that way ^[1]. The order 8 case was settled by a computer search ^[1].

For any even order 2m, it proved that a pairing always exists in which the number of blocks containing a repeat is at most (m + 3)/2 ^[1]. A count of blocks is a whole number, so for order 6, where m is 3, that means at most three blocks, and for order 8, where m is 4, the limit of three and a half also means at most three.

The same paper notes that the Latin squares of order 6 fall into twelve main classes, 22 isotopy classes and 17 transposition classes ^[1].

So those blocks are not the same as cages on a puzzle grid. The question is still a useful reminder that a square cage in a finished grid can hold a repeated digit on a diagonal, so never rule out a repeat in a square cage too early.

How do candidate lists connect to forbidden symbols?

Every time you cross a digit out of a cell, you are banning a symbol from that cell. Mathematicians ask what happens when many such bans are made at once.

A paper published on 12 May 2006 proved a general result about banned symbols in Latin squares: for any fixed number m, there is a size N such that whenever each cell of an N by N array bans at most m symbols, and no symbol is banned more than m times in any row or column, some Latin square avoids every ban ^[2].

That theorem says nothing about cage grids. Linking it to your candidate lists is only our own comparison, not a result of the paper.

What are the most common mistakes?

• Forgetting the order of subtraction and division. Take the larger digit first: 4 and 1 give 3 −, never a negative number.
• Repeating digits in a straight cage. A cage that lies along one row or column cannot hold the same digit twice.
• Ruling out repeats in bent cages. In an L-shaped or square cage, cells on different rows and columns may match.
• Ignoring the line total. In a four by four grid each line adds to 10, and in a six by six grid each line adds to 21. That check often finishes a cage.

What should you do when you get stuck?

• Find the cage with the shortest list of digit sets and see which sets the row and column rule now forbids.
• Pick a digit and ask where it can still go in each row and column, just as in a plain Latin square.
• Look for two cells in one line that between them must hold the same two digits. For example, if two cages that each need 3 and 4 both cross a row with one cell, those two cells take the 3 and the 4, so cross 3 and 4 out of every other cell in that row.
• Recheck any product cage that might include a 1, because a 1 leaves the product unchanged and is easy to overlook.

Can you solve this practice puzzle?

Fill the grid with 1 to 4, once in every row and column. Each letter marks a cage. This guide's own computer check found exactly one solution.

Hints. Cage q has three cells with a product of 2, so its digits are 1, 1 and 2. The two 1s cannot share a row or a column, so they sit in C1 and D2, and D1 is 2. Cages p, t and u each need 3 and 4, and cage s in column B needs 1 and 2. Use the columns to put each pair in order.

Answer:

If you enjoy reaching a target with a few operations, try this riddle next. It uses the same habit of testing each line against its total until only one value fits.