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Squares and tricks

65² is 6 × 7 then 25; 97 × 96 is 93 hundred and 12; 47 × 53 is 50² − 3². Mental squaring and near-base products, with numbers that keep growing.

  • Calculation
  • Timed
  • Endless levels

How it plays

Each question is a square or a product that has a shortcut. A number ending in 5: multiply the rest by one more and write 25. Near a base like 100: (100 + x)² = 10,000 + 200x + x². Two numbers near a base: add one's offset to the other, times the base, plus the offsets' product.

Numbers either side of a round one multiply as a difference of squares: (a + b)(a − b) = a² − b². Any square splits as (a + b)² = a² + 2ab + b².

Tricks unlock as you climb and the numbers grow without end. The wrong options are the trick misapplied — the +1 forgotten, the cross term dropped, the sign of the small product flipped — and the working shows the trick done right.

Keys

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

What it trains

This trains a handful of algebraic shortcuts for mental multiplication. Practice makes these faster; it claims nothing wider.

Learn

How to get better at squares and tricks: 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. Name the shape before you calculate: a square ending in 5, a square near a round number, two numbers near the same base, two numbers either side of a round one, or any other square.
  2. Pick the trick that shape calls for, and write the answer in two parts: a left part (hundreds, thousands or more) and a right part, the small product.
  3. Join the parts with the right number of digits on the right: two for a base of 100, three for 1,000. Pad with zeros if the small part is short; carry if it is long.
  4. With options, check the last digits first: the right part of the answer is usually all you need to tell the options apart.

Techniques and tricks

  • Ending in 5: times one more, then 25

    Drop the 5, multiply what is left by one more than itself, and write 25 after it.

    e.g. 65²: 6 × 7 = 42, so 4,225. 595²: 59 × 60 = 3,540, so 354,025.

  • Squares near 100 or 1,000

    Add the offset to the number (or take it off), then write the offset squared on the right, padded to two digits for 100, three for 1,000.

    e.g. 97²: 97 − 3 = 94, 3² = 09, so 9,409. 1,002²: 1,004 and 004, so 1,004,004.

  • Squares near 50 or 500

    Near 50, the hundreds are 25 plus the offset and the right part is the offset squared. Near 500, it is 250 thousand plus 1,000 times the offset, plus the offset squared.

    e.g. 53²: 25 + 3 = 28, 3² = 09, so 2,809. 507² = 250,000 + 7,000 + 49 = 257,049.

  • Two numbers near a base

    Add one number’s offset to the other number, multiply by the base, then add the product of the two offsets. Both below: the offsets’ product is still added.

    e.g. 97 × 96: 97 − 4 = 93, so 9,300, and (−3) × (−4) = 12: 9,312. 104 × 107 = 11,100 + 28 = 11,128.

  • Either side of a round number

    Two numbers the same distance either side of a round one multiply to its square minus the distance squared.

    e.g. 47 × 53 = 50² − 3² = 2,500 − 9 = 2,491. 408 × 392 = 160,000 − 64 = 159,936.

  • Split any square

    Split the number into a round part and the rest, and use (a + b)² = a² + 2ab + b². For trailing zeros, square the rest and write twice as many zeros.

    e.g. 43² = 1,600 + 240 + 9 = 1,849, so 430² = 184,900.

As the levels climb

  • The tricks unlock in order: squares ending in 5 at level 0, squares near 50, 100, 1,000 at level 1, products near a base and a² − b² together at level 4, and any square, split, at level 7. The newest is asked more often for a while.
  • Between unlocks, and for ever after, the numbers grow: more digits before the 5, bases of 1,000 and 10,000, bigger offsets, more zeros. The method never changes; only the bookkeeping of digits does.
  • The clock closes to 85% at level 3, 72% at level 6 and 62% at level 9. Write the left part as soon as you have it; the right part is a small product you can finish while choosing.
  • With bigger offsets, the offsets’ product can spill past the right part’s width. Then carry the extra digits into the left part.

Traps to avoid

  • The +1 forgotten on a 5: 65² given as 6 × 6 then 25, 3,625. It is always times one more.
  • The cross term halved or dropped: (b + x)² written as b² + bx + x², or a split square as a² + b². The middle term is 2ab, twice the product.
  • The small product’s sign flipped, or the offset squared taken away. When both numbers are below the base, (−3) × (−4) is +12 and is added.
  • One base out, from adding the wrong offset or misplacing a digit: 9,412 or 9,212 for 9,312. Check that the left part is the number plus the other’s offset.
  • For a² − b²: b² added instead of taken away, or b taken away instead of b². The answer is always a little less than the round number squared.

How to practise

  • Use the Tricks setting to drill one trick at a time until it is automatic, then switch them all back on to practise naming the shape.
  • Switch Answer by to “Typing the number”. The options differ mostly in the right part, so choosing lets you skip the left; typing makes you do both.
  • Say the two parts aloud as you go — “ninety-three hundred … and twelve” — so the joining step becomes a habit, not a calculation.

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.

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