Sum Triangle

Five numbers along the bottom, and every cell above them equal to the two cells directly below it added together. Seven of the fifteen are dealt to you and eight are holes; the eight missing numbers sit in a tray under the board, and your job is to put each one where it belongs.

The rule works in both directions, which is the whole point of the game. Forwards it is addition — a cell is the pair beneath it, summed. Backwards it is subtraction — a cell is the one above it take the cell beside it. Every board is dealt with the top cell missing and at least one bottom cell missing, so both readings are compulsory: the top has nothing above it and can only be added into place, a bottom cell has nothing below it and can only be subtracted into place. You cannot finish a board here by only ever adding.

The board tells you which holes are workable. A hole marked add can be read off the two cells under it; a hole marked minus can be read off the cell above it and the one beside it. A hole with no mark cannot be worked out yet — and noticing why is most of the skill.

A number that does not belong is refused rather than written, so the triangle can never be driven into a state with no answer. The only cost of a guess is one of your tries, and when they run out the round ends.

It runs entirely on your device: 0 network requests, works offline. No ads, no account, nothing to buy, and no betting, money or prizes of any kind.

1. Focus the board — click it, or press Tab until it is focused.
2. Move between holes with the left and right arrow keys, or A and D. The cursor only ever rests on a hole that is still open, so you never have to steer around cells you cannot write in. Moving costs nothing.
3. Jump between rows with the up and down arrow keys, or W and S. They land on the first open hole in the nearest row that still has one.
4. Hold a number from the tray with F (or Shift); Esc or Z steps back the other way. The tray is a ring, so one key reaches every number in it, and it is sorted smallest to largest. Choosing costs nothing either.
5. Press Enter or Space to put the held number into the hole under the cursor. If it belongs there it is written in, the tray loses it, and the cursor moves to the next open hole. If it does not, it is refused, the cell shows ✕ with the refused number under it, and you spend one try.
6. Or use the mouse: click a hole to move there, then click a number in the tray to put it in. One click is one move — clicking a cell never places anything, so you can never spend a try just by looking around.
7. Work the marked holes first. add means the two cells below it are both known, so their total goes here. minus means the cell above and the cell beside are both known, so the difference goes here. Fill one and the marks move — a hole that was blank a moment ago is usually marked now.
8. Fill all eight holes before the tries run out. Spare tries are worth points, and so is how much of the board had to be recovered backwards rather than added forwards.

How do you fill in a number pyramid where each cell is the sum of the two below it?

Never by trying numbers from the tray. Find a cell where two of the three numbers in one little triangle are already known, because that third one is then decided and no other number can go there. There are two shapes of that. If both cells below a hole are known, the hole is their total — that is the forward reading, and it is the one everybody finds first. If the cell above a hole and the cell beside it are both known, the hole is the one above take the one beside — that is the same rule rearranged, and it is the one that actually finishes boards. Write in whichever you find, then look again: the number you just wrote is now a known cell in two or three more little triangles, so it usually decides another hole immediately. Boards here collapse into a chain of those steps, and the game marks the workable holes for you so you can learn to spot them rather than be told where they are. The one thing never worth doing is starting from a number in the tray and hunting for a home for it.

Why is the top cell always missing?

Because it is the only cell that cannot be reached backwards. It has nothing above it, so no subtraction can ever produce it — the only way it is ever filled is by adding the two cells beneath it. Leaving it out guarantees that every board makes you read the rule forwards at least once. The mirror of that is at the bottom: at least one bottom-row cell is always missing too, and a bottom cell has nothing below it, so it can only be recovered by subtracting. Between those two, no board can be finished in one direction only. Neither is a lucky property of the deal — both fall out of the shape of the triangle, so they hold on every board without the generator having to check.

Is there always exactly one answer?

Yes, and two separate solvers say so before the board reaches you. Every triangle starts complete: the bottom row is drawn first and everything above it is computed, so the rule holds everywhere by construction and there is no such thing as a contradictory deal. Holes are then punched one at a time, and a hole is kept only if both solvers still agree. The first walks the forced chain — every cell it can decide from two known numbers, in waves, until nothing more moves — and if it finishes the board, the answer is unique and reachable without a single guess. The second ignores all of that and brute-forces every bottom row in range, builds each whole triangle and compares, stopping the moment it finds a second match. They share almost no code, so a disagreement would mean one of them is wrong; the tests check they never disagree, over thirty seeded boards and a set of hand-computed ones built to break each.

Why is a wrong number refused instead of written into the triangle?

Because being told twenty moves later is a punishment, not feedback. If wrong numbers could be written, a triangle could be driven into a state with no completion at all, and the only cure would be undoing a chain of moves you no longer remember making. Refusing the entry means every number on the board is part of the real answer, the puzzle can never dead-end, and the cost of a guess is one visible, finite try. It also changes what the game rewards: not careful transcription, but working out which hole is decided before you touch anything.

How many tries do I get, and what happens when they run out?

Between three and six, worked out from the board rather than picked. The forced-chain solver counts how many holes it could only reach backwards, and you get three tries plus one for every two of them — so a board that mostly adds upward gives you three, and a board that leans hard on subtraction gives you six. When they run out the round ends and the answer is written into the holes you never reached, marked answer, so you can see where the chain would have gone. Moving the cursor and changing which number you hold are both free and unlimited.

Is there any money, betting or in-app purchase in this?

None, and there never will be. No currency, nothing to buy, no hints to unlock for a fee, no prizes.

Does it work offline, and can I play it with a keyboard or a screen reader?

Yes to all three. Once the page has loaded once, the whole game — the board, the generator and both solvers that prove the answer is the only one — is on your device, and no server is involved in play at any point. Arrows or W A S D move, F or Shift holds the next number, Esc or Z the previous one, and Enter or Space puts it in. Every cell announces what it holds in words — dealt 12, your 7, empty and it is the sum of the two below it — and a refused entry says which number was refused, so nothing on this board depends on telling two colours apart.

Five rows, fifteen cells, eight holes, and a bottom row drawn from 1 to 9 — which puts the top cell somewhere between 16 and 144, and keeps every step inside two-digit mental arithmetic. There are 59 049 bottom rows and so 59 049 distinct triangles, and the deal punched into each one is different every time. Both solvers run exhaustively when the board is dealt rather than sampling, because at this size the brute-force count finishes in well under a millisecond — that is exactly what the five-row limit buys, and it is why every board can be proved unique rather than assumed to be. Boards are rated direct, chained or deep by how many waves the forced chain needed. There is no timer, no streak to protect and no daily limit.

Magic Square is the other grid where a wrong entry is refused the moment it is made, and where the move to look for is the line that is closest to forced. Sum Links and Pair Sums drill the same arithmetic from the other direction — given a total, find the numbers that make it — while Balance Scale works the same "two knowns decide a third" habit through weighings instead of sums. Gap Logic and Mini Sudoku run the identical do-every-forced-move-first discipline over an ordering and a grid.

Bookmark this page (Ctrl+D, or ⌘D on Mac) or install the app — it works offline the next time you need it.