In-Depth Tutorial
A formal, accessible 800–1000 word walkthrough of this topic, written for the serious aspirant. Switch to हिन्दी using the toggle on the right.
Why Squaring Is a Critical Speed Skill
Squaring numbers appears in almost every quantitative topic — areas of squares and circles, the Pythagorean theorem, standard deviation, statistics, time-distance problems involving relative speed, and the basic arithmetic of compound interest. A typical Banking Prelims paper contains four to six questions whose direct calculation requires squaring a two-digit number; a Staff Selection Commission Tier-I paper contains six to eight such questions. The conventional method of squaring 87 — multiplying 87 by 87 in long form — takes approximately twenty seconds. The Vedic method takes about six. Saving fourteen seconds on each of seven questions amounts to one minute and forty-three seconds per paper, almost three additional questions worth of time.
Vedic squaring rests on three central techniques: Ekadhikena Purvena for numbers ending in five, Yavadunam for numbers near a base, and the duplex method for any two-digit number whatever. The same hub also covers cubing, which is performed by Anurupyena, a proportional extension of the squaring rules.
Ekadhikena Purvena — Squaring Numbers Ending in Five
Numbers ending in five form a special family in which squaring is reduced to a single multiplication. The Sutra Ekadhikena Purvena instructs the candidate to multiply the leading part of the number by one more than itself, and to attach 25 on the right. Thus 25² is 2 × 3 = 6, attach 25, giving 625. Likewise 35² is 3 × 4 = 12 attach 25, giving 1225, and 75² is 7 × 8 = 56 attach 25, giving 5625.
The rule extends naturally to three-digit numbers ending in five. For 105², the leading part is 10; multiplied by 11 gives 110; attach 25 to obtain 11025. For 125², the leading part is 12; multiplied by 13 gives 156; attach 25 to obtain 15625. The technique is so simple that, with five minutes of practice, every aspirant can square any number ending in five — up to four digits — in under five seconds.
Yavadunam — Squaring Numbers Near a Base
Yavadunam, meaning 'whatever the deficiency,' applies when the number being squared is close to a base of ten, hundred or thousand. The two-step procedure is as follows. First, find the deficiency or surplus of the number from the base. Second, write the number minus the deficiency on the left and the square of the deficiency on the right, padded to the appropriate number of digits.
Consider 98². Base 100; deficiency 2. Left side 98 − 2 = 96; right side 2² = 04 (padded to two digits). Answer 9604. Consider 103². Base 100; surplus 3. Left side 103 + 3 = 106; right side 3² = 09. Answer 10609. Consider 992². Base 1000; deficiency 8. Left side 992 − 8 = 984; right side 8² = 064 (padded to three digits). Answer 984064. The Yavadunam method is dramatically faster than long multiplication for any number within ten of a power of ten, which covers a remarkably high proportion of competitive-examination questions.
The Duplex Method — Squaring Any Number
When the number to be squared is neither ending in five nor near a base, the duplex method delivers the answer in a single line. For a two-digit number ab, the duplex of the right column is b²; the duplex of the middle column is 2ab; the duplex of the left column is a². The candidate writes these three values in three columns and propagates carries from right to left.
Consider 47². Right column 7² = 49; middle column 2 × 4 × 7 = 56; left column 4² = 16. Writing 16, 56, 49 with carries: 49 contributes 9 to units and carries 4; 56 + 4 = 60 contributes 0 to tens and carries 6; 16 + 6 = 22 forms the leading digits. Answer 2209. The procedure extends to three-digit numbers with five columns following the standard duplex pattern. With twenty practice problems the candidate can square any two-digit number in under ten seconds and any three-digit number in under twenty.
Cubing — Anurupyena and the Proportionate Method
Cubing a two-digit number ab uses the Anurupyena rule. The candidate writes four columns: a³, a²b, ab², b³, with the middle two columns each multiplied by three before insertion. Carries propagate from right to left as in multiplication. Consider 12³. The four columns are 1, 2, 4, 8. Multiplying the middle two by three gives 1, 6, 12, 8. Writing with carries: 8 in units, 12 + 0 carries 1 → 2 in tens with carry, 6 + 1 = 7 in hundreds, 1 in thousands. Answer 1728.
For numbers near a base, the alternative cubing rule states that to cube a number near a base, write the number plus three times its surplus on the left, three times the surplus squared in the middle, and the surplus cubed on the right. For 102³: surplus 2; left = 102 + 6 = 108; middle = 3 × 4 = 12; right = 8. Padding the right to three digits and the middle to three digits: 108 (012) (008) → 1061208. The same logic, with appropriate sign treatment, works for numbers below the base.
A One-Week Mastery Plan
An efficient plan covers squares and cubes in seven days. Day one is dedicated to Ekadhikena with twenty squarings of numbers ending in five. Day two is Yavadunam with twenty squarings near a base. Day three is the duplex method with twenty general two-digit squarings. Day four extends the duplex method to three-digit numbers. Day five introduces Anurupyena for cubing two-digit numbers. Day six covers near-base cubing. Day seven is mixed practice with fifty assorted square and cube calculations under timed conditions.
By the end of the week, the candidate can square or cube any number commonly encountered in competitive examinations in under fifteen seconds with at least ninety-percent accuracy. The cumulative time saving across one paper is between two and three minutes — adequate for two extra confident attempts and a meaningful improvement in the final score.