Cage sums
Fill a Latin square so the numbers in each cage make its target by its sign — adding first, then all four operations. 4×4 up to 6×6.
How it plays
Every row and column holds each number from 1 up to the size once. Cells sharing a letter form a cage; its first cell shows the target and the sign, like 7+ or 2÷.
The cage's numbers must make the target by that sign: added, multiplied, or, for a pair, the larger minus the smaller or the larger divided by the smaller. A cage with no sign is a single cell holding that number.
Start where there are fewest ways — a one-cell cage, or 3+ in two cells, which can only be 1 and 2. Levels add operations, then grow the grid. Every puzzle has exactly one solution, checked by a solver.
Keys
Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.
What it trains
This practises arithmetic facts used backwards — which numbers make 12× — alongside elimination. It is practice at these puzzles, not a measured gain in reasoning elsewhere.
Learn
How to get better at cage sums: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.
Your approach to each question
- Fill the one-cell cages first. A label with no sign, like “D 3”, is the cell’s number — a free placement and a clue for its row and column.
- Then go cage by cage and list the sets of numbers that make each label, using only 1 to the grid size. Start with the cages that have one set, then those with two.
- Cross the sets with the rows and columns. A number already in a row cannot appear again in that row, which often cuts a cage from two sets to one.
- When a row or column has one cell left, finish it by the line total — 10 on a 4×4, 15 on a 5×5, 21 on a 6×6 — rather than by the cage.
Techniques and tricks
The extreme sums
In a two-cell cage the smallest sum, 3+, can only be 1 and 2, and the largest, one less than twice the size, can only be the two biggest numbers.
e.g. On a 4×4, 7+ is 3 and 4; on a 6×6, 11+ is 5 and 6
Factor the product
Break a × target into its factors and keep only the ways that use numbers up to the size. A prime like 5 in the target puts a 5 in the cage.
e.g. On a 6×6, 10× in two cells is 2 and 5 only; 12× is 2 and 6, or 3 and 4
Divide and differ
For a ÷ pair, list each small number with its multiple: on a 6×6, 2÷ is 1 and 2, 2 and 4, or 3 and 6. For a − pair, the biggest gap — one less than the size — can only be 1 with the top number.
e.g. On a 5×5, 4− is 1 and 5; on a 6×6, 3÷ is 1 and 3, or 2 and 6
The line total
Every row and column adds up to the same total. When a cage sits wholly inside a line, subtract it to learn what the rest of the line must add to.
e.g. On a 5×5 a row is 15; a 7+ cage across three of its cells leaves 8 for the other two
Repeats in a bent cage
A number may repeat inside a cage when the two cells are in different rows and columns, as at the two ends of an L. Do not throw out a set just because it repeats.
e.g. On a 4×4, 16× in an L of three: 4 × 4 × 1 or 2 × 2 × 4 — either way the two ends of the L hold the same number
The locked pair
When two cells in one row must be the same two numbers, between them they use up those numbers for that row: no other cell in it can take either.
e.g. A 3+ pair lying across a 4×4 row puts 1 and 2 there, so the row’s other two cells are 3 and 4
As the levels climb
- The ladder starts on a 4×4 with adding only, then brings in subtraction, then all four operations, with cages of up to three cells. Learn the sets for each new sign on the small grid, where there are few.
- Next comes a 5×5 with adding only, then with all four, then with cages of up to four cells. Bigger cages have more sets, so lean on the line total and on crossing rows and columns rather than on listing alone.
- The top three rungs are 6×6: add and subtract, then all four, then cages of up to four. The 6 makes many labels two-way — 6× is 1 and 6 or 2 and 3, 12× is 2 and 6 or 3 and 4, 3÷ is 1 and 3 or 2 and 6 — so Factor the product needs care.
- With “Operations” on “Added by level”, the signs come in as above. “Adding only” or “All four” overrides the sign at every rung, while the grid still grows.
- The clock is off unless you turn on Time pressure; then a grid gets 90 seconds plus 12 seconds a cell, so nearly five minutes for a 4×4 and nearly nine for a 6×6.
Traps to avoid
- Doing − or ÷ the wrong way round. The label is always the larger number minus, or divided by, the smaller, so 2÷ allows 4 above 2 and 2 above 4 alike. The order inside the cage is free.
- Keeping a pair that cannot be. A two-cell cage always shares a row or a column, so its two numbers must differ: on a 5×5, 4× in two cells is only 1 and 4, never 2 and 2.
- Rejecting a repeat that is allowed — or allowing one that is not. Two cells of a cage may share a number only if they share neither a row nor a column.
- Reading the letter as part of the label. “B 7+” is cage B with target 7 and the sign +; the other cells of cage B show only “B”.
How to practise
- Set “Operations” to “Adding only” until the sums of two and three cells come without thinking, then switch to “All four” to drill products and quotients.
- Set “Largest grid” to 4 while you learn the signs, and raise it to 5, then 6, once a 4×4 with all four operations falls without a guess.
- Before you place a single number, work out the sets for the two or three tightest cages. That habit is most of the skill.
- Leave Time pressure off while learning; turn it to Relaxed once you can finish a 5×5, to make the set-listing quick.
Levels
It is untimed unless you turn the clock on. There are 9 levels with the default settings, and the counter stops at the real top. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.
All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.
More logic
All games →Name the fallacy
Ad hominem, straw man, false dilemma, post hoc, slippery slope: read a short argument and name the one move it makes. Fresh wording each time.Bridges
Join numbered islands with straight bridges, one or two per pair, never crossing, all in one group. Small boards first; every puzzle has one solution.Who is tallest?
From shuffled comparisons — Ana is taller than Ben, Cleo is shorter than Ana — build the line in your head and say who stands where.Off by one
Fence posts, cuts, floors, inclusive pages, a striking clock, lists numbered from zero — count the ends right, every time, at speed.Knights and knaves
Everyone on the island always tells the truth or always lies. From what they say, work out who is which — more islanders, harder sentences.Syllogisms
All wugs are daxes; some daxes are feps. Does it follow that some wugs are feps? Judge conclusions about made-up kinds, from two premises to long chains.