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Quick estimate

Which is bigger, 38 × 21 or 29 × 27? Judge size fast, without working it out — the two sides creep closer every level and never meet.

  • Number sense
  • Timed
  • Endless levels

How it plays

Two products: which is bigger? Or a product and four round numbers: which is closest? Or a board of ● and ○ flashed for a moment: which was there more of? There is not time to calculate exactly; that is the point.

Levels bring the two quantities closer together — 60% apart, then 49%, then 40%, on toward 0 without arriving — and make the numbers longer and the time shorter.

After each answer you see the exact values and how far apart they really were.

Keys

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

What it trains

This practises judging magnitude quickly. Estimation improves with practice at estimation; that is all this page claims.

Learn

How to get better at quick estimate: 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. Count the digits of each product first: a product of a 3-digit and a 2-digit number is a 4- or 5-digit number. Early on, a difference in digits settles it at once.
  2. If the sizes match, compare factor by factor rather than multiplying anything out: how much bigger is A’s first number than B’s, and how much smaller is its second?
  3. For “closest”, round each number to two figures, multiply, and count the zeros. The options are spaced far enough apart that this is all you need.
  4. Answer, then read the working: it gives the exact values and how far apart they really were. That percentage is the measure of how fine your judgement has to be.

Techniques and tricks

  • Compare the ratios, not the products

    A × B against C × D: A is bigger whenever A ÷ C beats D ÷ B. Two divisions of similar numbers are far easier than two multiplications.

    e.g. 38 × 21 or 29 × 27: 38 ÷ 29 ≈ 1.31 and 27 ÷ 21 ≈ 1.29, so 38 × 21 wins, just (798 against 783)

  • Biggest factor is not biggest product

    Before you trust the largest number on the screen, check what its partner is. A big number times a small one often loses.

    e.g. 664 × 10 against 982 × 5: 982 is the biggest number, but 6,640 beats 4,910

  • Small percentages add

    When the factors differ by a few per cent, add the per cents: up 6% and down 4% is about up 2%. It is close enough once the gap is under about 20%.

    e.g. 53 × 48 against 50 × 50: +6% and −4%, about +2%. Exactly 2,544 against 2,500, 1.8% more.

  • Equal up and down loses a little

    One factor up by x% and the other down by the same x% always gives a slightly smaller product, never the same one.

    e.g. 52 × 48 = 2,496, just under 50 × 50 = 2,500

  • Two figures and the zeros

    For “closest”, keep the first two figures of each number, multiply those, then put the zeros back. Rounding to one figure is too coarse when a number starts with 1.

    e.g. 664 × 102 ≈ 660 × 100 = 66,000, so 68,000 is the closest of 1,100 · 8,700 · 68,000 · 530,000

  • Count the rarer symbol, then stop counting

    On a small flashed board, count the symbol there is less of in twos and threes; the other must be the rest. Once the board grows past what you can count in a glance, take in the whole board and judge which one fills more of it.

As the levels climb

  • The two quantities close in every few levels — 60% apart, then 49%, then 40%, and on towards 0 without ever meeting. Under about 20%, digit counts and one-figure rounding stop working; switch to comparing ratios.
  • The numbers grow a digit at a time, alternating between the two factors. Count each product’s digits before anything else, because the factors of A and B do not always have the same lengths.
  • The clock starts at 4.5 seconds plus 0.6 for each step of size, and closes to 55%. Skip the multiplication entirely: two quick divisions or a per-cent sum is all there is time for.
  • On the flashed board, the count grows by three dots a step and the board shows for less time, from 1.3 seconds down to about a third of a second. Fix your eyes on the centre before it appears so the whole board lands at once.

Traps to avoid

  • Trusting the biggest single number. 982 × 5 looks bigger than 664 × 10 because of the 982; it is not. Always weigh a number against its partner.
  • Taking an equal up and down as a tie. 52 × 48 and 50 × 50 are not equal; the one with the spread is smaller, by the square of the half-gap.
  • Rounding both numbers the same way when they are close. 66 × 10 and 69 × 10 both round to 70 × 10; when the gap is small, the rounding has to be finer than the gap.
  • Counting one corner of the flashed board and guessing the rest. The symbols are scattered at random, so a corner can easily lean the wrong way.

How to practise

  • Start on “Which product is bigger?”, the default. Add “Which round number is closest?” to practise placing a single product, and use “A mix of all three” once both feel steady.
  • Play “More ● or ○? (no numbers)” for a few rounds to practise comparing amounts by eye. It is the same judgement of more and less, with no digits to lean on.
  • Keep an eye on the “% apart” in the level name. When you start missing, note that percentage: it is the finest gap you can currently judge, and the thing to push down.

Levels

Each question has a clock that comes from the work it asks for. 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 number sense

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