gamesblog

Magic square

Every row, column and diagonal adds to the same sum. Fill the marked gap — 3×3 up to 7×7, more blanks, bigger numbers, without end.

  • Calculation
  • Untimed
  • Endless levels

How it plays

A square of numbers in which every row, every column and both diagonals have the same sum. Some cells are blank; one is marked ?. A line with only one blank gives it away: the sum minus the rest.

Levels grow the square from 3×3 to 4×4, 5×5 and 7×7, add blanks so the ? needs a chain of lines, and make the numbers bigger without limit. Turn off the stated sum and you must first read it from a complete line.

Every puzzle is checked to be reachable line by line before it is shown, so the answer is forced. The working shows the chain.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This practises quick addition and subtraction and choosing which line to use first. It is practice at these puzzles, nothing wider.

Learn

How to get better at magic square: 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

  1. Get the sum first. It is stated in the question unless you turned that off; then add up any complete row, column or diagonal.
  2. Look at every line through the ? — its row, its column and, if it is on one, a diagonal. If one of them has no other blank, the ? is the sum minus the rest.
  3. If every line through the ? has another blank, find a line with exactly one blank that fills one of those cells. Fill it, then come back to the ?.
  4. Add the known numbers in the line first and subtract once at the end. One subtraction is much safer than four in a row.

Techniques and tricks

  • Sum minus the rest, in one go

    Total the numbers you can see in the line, then take that total from the sum.

    e.g. Sum 21, row 6, 11, ?: 6 + 11 = 17, and 21 − 17 = 4.

  • The 3×3 centre is a third

    In every 3×3 magic square the centre is the sum divided by 3. If you know one, you know the other.

    e.g. Sum 21, so the centre is 7; a centre of 6 means a sum of 18.

  • Opposite through the centre

    In a 3×3, two cells opposite each other through the centre add to twice the centre. A corner gives the far corner; an edge gives the far edge.

    e.g. Centre 7: the cell opposite a 10 is 14 − 10 = 4.

  • The 4×4 corners and quarters

    In the 4×4 squares this game builds, the four corners add to the sum, the middle four add to the sum, and so does each 2×2 quarter. These give you lines the rows and columns do not.

  • Diagonals that wrap round

    The 5×5 and 7×7 squares here are built so that a diagonal running off one edge and coming back on the opposite edge also adds to the sum. When the main lines are blocked, try one of these.

  • Size check

    A line of n cells adds to the sum, so its cells average sum ÷ n. An answer far from that average, or negative, has gone wrong.

As the levels climb

  • The square grows first: 3×3 at level 0, 4×4 at level 1, 5×5 by level 4. 7×7 only comes if you pick it under “Largest square”; the default stops at 5×5.
  • The numbers grow without end, taking turns with the size and the blanks. Bigger numbers change nothing in the method, but make the add-then-subtract-once habit matter.
  • Blanks are added four times, up to 9 on a 5×5 by level 10. With more blanks the ? needs a chain: one line to fill a cell, another to reach the ?. The working shows that chain.
  • On 4×4 and larger, the corner, quarter and wrap-round lines become the quickest way to a cell the rows cannot reach.

Traps to avoid

  • One number not subtracted. The answer comes out too big by exactly one of the line’s numbers. Count the cells you have added: it should be the line’s length less one.
  • The total of the rest given instead of the gap: 17 picked when the sum is 21 and the ? is 4. You stopped one step early.
  • One out or ten out, from a slip in the adding. Check the last digit of your total against the last digits of the numbers.

How to practise

  • Turn off “State the sum” once the puzzles feel easy. Finding a complete line first is a separate skill and makes you look at the whole square.
  • Set “Largest square” to 7×7 when 5×5 is comfortable; the bigger square makes the wrap-round diagonals worth knowing.
  • Switch Answer by to “Typing the number”. Options let you pick the one near the average; typing makes you do the line.
  • Leave Time pressure Off and plan the chain before you add anything: say which line fills which cell, then compute.

Levels

It is untimed unless you turn the clock on. There is no last level: at least one dial turns without end. 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 calculation

All games →