Mini Sums: How to Think Through a 5x5 Cage Puzzle

The rules take two lines. Fill the grid so every row and every column holds the numbers 1 to 5 exactly once. Then make every outlined cage produce the number in its corner, using the sign drawn on the line between its cells.

That second line is what makes people reach for a pencil and start adding. It is the wrong reflex. Mini Sums is a deduction puzzle that happens to look like arithmetic, and the arithmetic you need is small enough to hold in your head.

The cages that can only be one thing

With five numbers in play, most cages have fewer answers than they look like they should.

Start with a two-cell times cage. Every pair of different numbers from 1 to 5 multiplies to a different result. 2, 3, 4, 5, 6, 8, 10, 12, 15 and 20 each come from exactly one pair, so a 6 with a times sign is 2 and 3, full stop, and a 12 is 3 and 4. The only thing left open is which cell takes which, and the row or column usually settles that.

Divide cages are nearly as tight. A 3, 4 or 5 with a divide sign has to include a 1. Only a 2 with a divide sign leaves you a real choice, between 1 and 2 or 2 and 4.

Minus cages are the loose ones, and a 1 with a minus sign is the least helpful clue on the board: it could be any neighbouring pair, and it is usually the last cage you settle in that corner of the grid. A 4 with a minus sign, on the other hand, is 1 and 5.

Plus cages sit in between. In two cells, 3, 4, 8 and 9 have one answer each, while 5, 6 and 7 have two. Three-cell times cages get tight again at the top end. 60 is 3, 4 and 5 in some order, and 40 is 2, 4 and 5.

Before you place anything, walk the grid once and mark, in your head or with notes, the cages that are already decided. On most days three or four are. They are your foothold.

A number can repeat inside a bent cage

One rule catches newcomers. The once-per-row-and-column rule is the only repeat rule there is. A cage that bends around a corner has two cells that share neither a row nor a column, and those two cells may hold the same number.

This matters more than it sounds. An 11 with a plus sign across three cells can only be 2, 4 and 5 if all three numbers differ, so when that spread will not fit, look for a repeated pair, such as 5, 1, 5 with the two 5s on the diagonal. A 25 with a times sign in an L-shape has to be 5, 1, 5. If a cage seems impossible, check whether it is bent, and let a number repeat.

The row rule does half the work

Every number appears exactly once in each row and once in each column. On a 5x5 grid that is a tight constraint, and it is the reason a placed number travels so far.

Once you have filled a cage, look along its row and down its column before you look at the next cage. A row with three numbers has two cells left and two numbers left, and the cages those cells belong to almost always decide which goes where. The same goes for a column with three numbers. Fill the fullest line first, always.

The rule sharpens the cages too. A 5 with a plus sign in two cells is 1 and 4 or 2 and 3, but if the cage sits in a row that already holds a 4, it is 2 and 3. A lot of the game is a cage offering two answers and a line choosing one.

Watch the sign, not just the number

Two cages can carry the same number and mean completely different things. A 2 with a divide sign and a 2 with a minus sign look alike from across the grid and share no answers. On the weekend boards that is deliberate. The signs are chosen so that the number alone tells you less, which puts the information on the line between the cells.

So read the sign before you read the number, every time. It sits on the line between two cells of the same cage rather than in a corner, and your eye lands on it as you trace the cage.

When the grid stops moving

Somewhere in the middle of a solve the decided cages run out and nothing obvious is left. The board is working as intended, and there is always a next move. Two habits find it.

The first is to hunt for a number rather than a cell. Pick a number that is already placed three times and ask where it can go in the two rows that lack it. The row rule and the column rule between them usually leave a single cell, and that cell often sits in a cage you had given up on.

The second is to pin a cage to a line. If a two-cell cage lies entirely inside one row and has to contain a 5, then no other cell in that row can be 5, even before you know which of the two cells takes it. Pencil the 5 into both cage cells and strike it from the rest of the row. On Saturday and Sunday this move opens the board more often than any other.

Notes help here. Switch them on for the cage cells that are down to two options and leave the rest of the grid clean. Pencilling every cell on a 5x5 mostly creates a mess you then have to maintain.

About the colours

When a cage is full, its corner number turns green if the arithmetic works and red if it does not. A repeated number in a row or column turns red the moment you place it. They catch slips. Using them to explore is guessing with extra steps. Every board can be finished by reasoning alone, so if you catch yourself trying numbers and watching for red, the move you want exists and you have not found it yet. Step back, pick a different number to hunt, and come at it again.

What changes as the week goes on

The grid stays 5x5 all week and the numbers stay 1 to 5. What changes is how much you are given and how much the signs tell you. Monday to Friday, a few cells start filled and most cages are the decided kind. Saturday and Sunday, there is one fewer given, and the signs are picked so that more cages hold two answers instead of one, which means more pinning and more hunting before the board falls. The day-by-day picture, including how long each day should take, is in how the daily week works.

Mini Sums is inspired by KenKen, the arithmetic grid puzzle created by Tetsuya Miyamoto. If you want bigger grids with the same idea, the practice library has KenKen boards from 3x3 up to 6x6.

Frequently asked questions

Do the cage numbers have to be different?

No. Only rows and columns forbid repeats. A cage that bends around a corner can hold the same number twice, as long as the two cells share no row or column.

Why 1 to 5 and not 1 to 9?

Five numbers keep the arithmetic in your head and the deductions fast. A 5x5 board takes most people five to eight minutes, which is about right for something you play every day.

What if I get stuck?

There is always a findable next move. Hunt for a number instead of staring at a cell, and check the fullest row and column. If that turns up nothing, look for a cage that pins a number to a line.

Can I ever be forced to guess?

No. Every board is checked by a separate program before publication to prove it has exactly one solution reachable by logic. If a board needed a guess, it never reached you.