Cage Arithmetic Tips: Solve KenKen Cages with Row and Column Logic (2026)

1. What you are actually playing

Cage Arithmetic is a 4 by 4 kenken puzzle. There are no given numbers: every one of the 16 cells starts empty, and the only clues are the cages drawn on the grid. Each cage prints an operation and a target, and the digits inside it must combine to that target. On top of that, every row and every column must contain 1, 2, 3 and 4 exactly once. That single extra rule — row and column uniqueness, with no boxes — is what makes a cage puzzle solvable by logic rather than guessing.

There is no timer, no score and no lives. A puzzle is finished when all 16 cells are filled and every cage, row and column is satisfied, at which point the game shows "Solved — all cages correct." and you can move to the next or previous puzzle. Three hints are available per puzzle if you get stuck, and undo lets you backtrack as far as you need.

2. Reading a cage

Each cage is a group of one or more cells with a thick border around them. Its clue sits in the top-left cell of the cage and looks like one of these:

• A bare number, such as 3= written as just 3 with op =: that cell must contain exactly that digit. Single-cell cages are free givens.
• 7+: the cells in that cage must add up to 7.
• 24* (shown as 24×): the cells must multiply to 24.
• 2-: the two cells must differ by 2 (absolute value, order does not matter).
• 3÷ (shown as 3÷): the larger of the two cells divided by the smaller must equal 3.

Two important rules that are easy to get wrong:

• Kenken does not require the digits inside one cage to be distinct. A cage may repeat a digit as long as that digit does not repeat in its row or column. For example a 4+ cage of two cells can be 2 and 2.
• Subtraction and division cages always have exactly two cells in this build. Addition, multiplication and single-value cages can have any size.

The partial-placement check in the engine also helps while you play: if a + cage already exceeds its target, or a * cage already exceeds its target, the cell you just typed turns red. Use that as a live filter, not as a solution.

3. The tight cages (your starting points)

On a 4 by 4 grid the digits are only 1, 2, 3, 4, so every cage has a very short list of possible fills. Learn these by heart; they are the engine of every solve.

Single cell: the digit is given.

Two cells:

• 3-: only (4, 1) in either order.
• 3÷: only (3, 1) in either order.
• 1-: (2,1), (3,2), (4,3).
• 2-: (3,1), (4,2).
• 1÷: (2,2) — wait, that repeats; actually (2,1)? No: 2/1=2, 4/2=2, 3/? no. Concretely: quotient 2 needs pairs (4,2) or (2,1); quotient 3 needs (3,1); quotient 4 needs (4,1).
• 3+: (1,2) only (two distinct? could be (1,2); (??) (??) no other positive pair sums to 3 with 1-4 except 1+2).
• 7+: (3,4) only.
• 5+: (1,4) or (2,3).
• 4+: (1,3) or (2,2).
• 6+: (2,4) or (3,3).
• 8+: (4,4) (repeat, allowed if rows/cols permit).
• 2*: (1,2).
• 3*: (1,3).
• 4*: (1,4) or (2,2).
• 6*: (2,3).
• 8*: (2,4).
• 12*: (3,4).
• 16*: (4,4).

Three cells:

• 6+ of three cells: only 1 + 2 + 3.
• 7+ of three cells: 1 + 2 + 4.
• 8+ of three cells: 1 + 3 + 4 (or 2 + 2 + 4, or 2 + 3 + 3).
• 9+ of three cells: 2 + 3 + 4 (or 1 + 4 + 4, or 2 + 2 + 5 impossible).
• 6* of three cells: 1 * 2 * 3.
• 8* of three cells: 1 * 2 * 4.
• 12* of three cells: 1 * 3 * 4.
• 24* of three cells: 2 * 3 * 4.

Four cells:

• 10+ of four cells: 1 + 2 + 3 + 4.
• 24* of four cells: 1 * 2 * 3 * 4.

The moment you see one of these tight cages, write its digits in (or note the two/three candidates) before touching anything else. That is the whole game.

4. A solve order that works

1. Scan for single-cell cages and write them in. They are free.
2. Scan for two-cell cages with a unique pair (3-, 3÷, 7+, 12*, etc.) and write both cells in when their row and column allow it.
3. Scan for three- and four-cell cages with a unique set and write them in.
4. For cages with two candidate sets, use row and column uniqueness to decide. If a cage needs a 4 and its row already has a 4 elsewhere, the cage cannot use 4.
5. After each write, re-scan: a filled cell removes that digit from its row, column and (loosely) its cage.
6. When nothing is forced, look for a row or column with three filled cells — the fourth is forced by uniqueness even if its cage is not.
7. Only guess when every cage and every row/column is ambiguous; use undo to backtrack, or spend a hint.

The engine's red highlight on partial cage overflow is a useful guard during step 4-6: if a candidate makes a + cage exceed its target, discard it.

5. Using the 3 hints well

Hints are not "solve the puzzle" buttons. Each hint fills exactly one cell — the first empty cell in row-major order — with the correct digit, or, if no cell is empty, corrects the first cell that is filled but wrong. Three consequences:

• A hint can land on a cell you have not even looked at yet, which may not be the one you wanted. Spend hints late, when the puzzle is almost done and you need a tie-breaker.
• A hint counts as a move for undo: the previous board is pushed onto the history, so you can undo a hint if it did not help the way you hoped.
• Hints do not skip a puzzle. After three hints you are on your own for the rest; use undo and restart freely.

Practical rule: try to solve without hints first. If you are stuck on the last two or three cells, one hint usually cracks the puzzle because it removes the last ambiguity.

6. Common traps

• Treating kenken like sudoku. There are no boxes; the only uniqueness is row and column. A digit may appear twice in the same cage if the rows and columns allow it.
• Forgetting that subtraction and division are absolute. 2- means the two cells differ by 2 in either order; 3÷ means the larger divided by the smaller is 3.
• Over-trusting the red highlight. It only catches overflow on + and * cages; it does not catch a wrong digit that still fits the cage. Verify against row and column uniqueness.
• Filling a cage's cells without checking the row. A 4* two-cell cage could be (1,4) or (2,2); if the row already has a 2, (2,2) is impossible.
• Burning hints early. Three hints on a 16-cell puzzle is a lot; save at least one for the endgame.

7. Difficulty tiers

• Easy (e-1 to e-3): small cages, many single-cell givens and unique pairs. Good for learning the tight-cage tables in section 3.
• Medium (m-4 to m-6): larger cages, more two-candidate sets, and at least one cage that needs row/column reasoning to resolve.
• Hard (h-7 to h-9): cages span full rows or columns, product cages with multiple factorisations, and puzzles where the last cells need a chain of implications.

All nine puzzles are frozen, so a puzzle you restart is identical. The prev/next buttons (or the arrow keys on the grid) move between puzzles in the current tier; the tier buttons switch tiers. There is no scoring, so the only thing to beat is your own time and hint count.

8. Keyboard and touch

• Tap or click an empty cell to select it; type 1-4 to fill, Backspace/Delete/0 to clear.
• Arrow keys move the selection cell by cell (skipping given cells).
• The on-screen digit pad, erase, undo, hint, restart, prev and next buttons do the same on touch devices.
• The grid resizes to fit the screen; play in landscape on a phone for the largest cells.

The rules above are strategy; the game page already covers moving, filling and restarting. Every constant here was read from the shipped build, so if a future update changes a number, this guide changes with it.