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Count the squares

How many squares of every size are in a grid — or how many rectangles? Count by the formula, not by staring, as the grids grow without end.

  • Spatial
  • Untimed
  • Endless levels

How it plays

A grid of small squares hides bigger ones: a 2 × 2 grid has four small squares and one big one, five in all. Count every square of every size — or, in the other kind of question, every rectangle.

Squares: a k × k square fits (n − k + 1)(m − k + 1) ways, so add that up for every k. Rectangles: each is two of the horizontal lines and two of the vertical lines, so multiply the ways to choose them. The working shows both every time.

Levels grow the grid without end and stretch it from square to oblong. Answer by choosing, or type the count.

Keys

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

What it trains

This practises counting by structure — splitting a count by size and multiplying choices — on grids like these. It claims nothing wider.

Learn

How to get better at count the squares: 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. Read the question first: squares, or rectangles (squares included). They use different formulas, and each is a wrong option for the other.
  2. Read the grid’s size as rows × columns of small squares. Count the squares along one edge, then the other; big grids are given in words, “A grid of 23 × 18 small squares”.
  3. Never count by staring. Squares: add one product for each size of square. Rectangles: pick two lines each way and multiply.
  4. Before you pick, check the size of your answer against the options. The wrong ones are the counts a slip gives, so they are close to plausible — your formula decides, not your eye.

Techniques and tricks

  • One size at a time

    In an n × m grid (n the shorter side), a k × k square fits (n − k + 1)(m − k + 1) ways. Write the products for k = 1, 2, 3… and stop when one side reaches 1.

    e.g. 3 × 5: 3 · 5 + 2 · 4 + 1 · 3 = 15 + 8 + 3 = 26 squares

  • Square grids: sum of squares

    On an n × n grid the products are n², (n − 1)², … 1², which add to n(n + 1)(2n + 1) ÷ 6.

    e.g. 8 × 8: 8 · 9 · 17 ÷ 6 = 204 squares

  • Oblong grids in one line

    For an n × m grid with n ≤ m, every size added up comes to n(n + 1)(3m − n + 1) ÷ 6. Handy once the grid is too big to list every size.

    e.g. 4 × 6: 4 · 5 · 15 ÷ 6 = 50 squares

  • Rectangles are two pairs of lines

    A rectangle is fixed by its top and bottom lines and its left and right lines. An n × m grid has n + 1 lines one way and m + 1 the other. Choose 2 of each and multiply.

    e.g. 3 × 4: 4 lines give 6 pairs, 5 lines give 10 pairs, so 6 × 10 = 60 rectangles

  • Triangular numbers

    Choosing 2 lines from k + 1 is 1 + 2 + … + k = k(k + 1) ÷ 2. Learn them up to about 20 (1, 3, 6, 10, 15, 21, 28, 36, 45, 55…) and rectangles become one multiplication.

    e.g. 6 × 10 grid: 21 × 55 = 1,155 rectangles

  • Last digit decides

    The slips often land close to the truth in size. Work out only the last digit of your sum or product first; it usually rules out all but one option.

    e.g. 11 × 6 grid, options 196, 1,386, 990, 87: 21 × 66 must end in 6, so only 196 and 1,386 survive, and 196 is far too small

As the levels climb

  • Levels take turns: one grows the grid a row and a column, the next stretches it a column longer, four times over; after that it only grows. Expect oblong grids, which the squares count handles differently: the sizes stop at the shorter side.
  • Past 12 squares a side the grid is no longer drawn — only its size is given. That is the point of the formulas: once you trust them, a 23 × 18 grid is no harder than a 3 × 4.
  • The time grows with the grid, 2.5 seconds for each square on its short side. The game is untimed unless you turn on Time pressure; if you do, the one-line formulas save the most time on big grids.

Traps to avoid

  • Only the small squares: the answer rows × columns. That is just the k = 1 term — every bigger size is missing.
  • The biggest squares left out, or one whole size dropped. List every k from 1 to the short side and check you have that many terms.
  • Treating an oblong grid as square: using the sum of squares n²+…+1² on a 3 × 5 grid. Only a square grid sums squares; an oblong one multiplies two different numbers each time.
  • Rectangles counted from the squares instead of the lines: 2 of 3 and 2 of 4 on a 3 × 4 grid, 3 × 6 = 18, instead of 2 of 4 and 2 of 5. Or one side right and the other short by one. A 3 × 4 grid has 4 and 5 lines.
  • Adding the two choices of lines when they should be multiplied: 6 + 10 instead of 6 × 10. Every pair of horizontal lines goes with every pair of vertical ones.

How to practise

  • Start with only Squares ticked under Count, then only Rectangles, until each formula comes without thinking; then tick both, so you have to read which one is asked.
  • Set Answer by to “Typing the number” once the options feel easy. The slips sit among the options, and typing stops you choosing by look.
  • Raise Options shown towards 8 to bring in more of the near-miss counts, so only an exact answer gets through.
  • Check one small grid by hand, once: a 2 × 3 has 6 + 2 = 8 squares and 3 × 6 = 18 rectangles. Seeing the formula agree is what lets you trust it on a grid too big to draw.

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.

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