Cubing a number means multiplying it by itself three times, so 4 cubed, written 4³, is 4 × 4 × 4, which is 64. Done the ordinary way, cubing a two-digit number takes two long multiplications. Vedic Maths has two shortcuts that cut this to a few small steps: Anurupyena, which works for any two-digit number, and Yavadunam, which is fastest for numbers close to 10, 100 or 1,000. This guide explains both step by step, shows why they work, and ends with a practice set.
Both methods use small cubes, so it helps to know these by heart:
| Number | Cube | Number | Cube |
|---|---|---|---|
| 1 | 1 | 6 | 216 |
| 2 | 8 | 7 | 343 |
| 3 | 27 | 8 | 512 |
| 4 | 64 | 9 | 729 |
| 5 | 125 | 10 | 1,000 |
Notice the last digits. Every cube from 1 to 10 ends in a different digit: 2 cubed ends in 8, 3 cubed ends in 7, 7 cubed ends in 3 and 8 cubed ends in 2, while the others end in themselves. That pattern is what makes Vedic cube roots so quick.
Anurupyena means "proportionately". For a two-digit number, call the tens digit a and the ones digit b. The method builds a row of four numbers that grow in the same proportion, then adds a second row underneath.
Here a is 1 and b is 2, so each number in the row is twice the one before: 1, 2, 4, 8. Under the middle two, write their doubles, 4 and 8. The column totals are 1, then 2 + 4 = 6, then 4 + 8 = 12, then 8.
From the right: write 8. Then 12: write 2 and carry 1. Then 6 + 1 = 7. Then 1. The answer is 1,728.
Here a is 3 and b is 4. The row is 27, 36, 48 and 64: each one is the one before times 4, divided by 3. The doubles of the middle two are 72 and 96, so the column totals are 27, then 36 + 72 = 108, then 48 + 96 = 144, then 64.
From the right: 64 gives 4, carry 6. Then 144 + 6 = 150: write 0, carry 15. Then 108 + 15 = 123: write 3, carry 12. Then 27 + 12 = 39. The answer is 39,304.
The row is 8, 20, 50 and 125, and the doubles are 40 and 100. The columns are 8, 60, 150 and 125. From the right: write 5, carry 12. Then 150 + 12 = 162: write 2, carry 16. Then 60 + 16 = 76: write 6, carry 7. Then 8 + 7 = 15. The answer is 15,625.
Why it works: multiplied out, (a + b)³ is a³ + 3a²b + 3ab² + b³. The row gives a²b and ab² once, and the doubles add them twice more, which makes three of each. Writing the terms in columns takes care of the place values.
For a number close to 10, 100 or 1,000, find how far it is from the base. Call that difference d: positive above the base, negative below. The answer has three parts:
The middle and right parts each get as many digits as the base has zeros.
The base is 100 and 104 is 4 above it. The left part is 104 + 8 = 112. The middle part is 3 × 16 = 48. The right part is 4 × 4 × 4 = 64. Side by side, the answer is 1,124,864.
Here 98 is 2 below 100, so the difference is negative. The left part is 98 − 4 = 94. The middle part is 3 × 4 = 12, because squaring makes the difference positive. The right part is the cube of minus 2, which is minus 8. Write the first two parts side by side to get 941,200, then take away the 8. The answer is 941,192.
The base is 1,000 and 997 is 3 below it. The left part is 997 − 6 = 991. The middle part is 3 × 9 = 27, written as 027. The right part is minus 27. Side by side, the first two parts give 991,027,000, and taking away 27 leaves 991,026,973.
Why it works: for a number d away from the base, the cube multiplies out to the base squared times (the number + 2d), plus the base times 3d², plus d³. Those are exactly the three parts. The same base idea is used for squares in Vedic Maths square tricks.
Two quick checks catch most slips:
Answers: (1) 2,197: the row is 1, 3, 9 and 27, and the columns are 1, 9, 27 and 27. (2) 9,261: the row is 8, 4, 2 and 1, and the columns are 8, 12, 6 and 1. (3) 91,125: the columns are 64, 240, 300 and 125. (4) 1,092,727: the parts are 109, 27 and 27. (5) 884,736: the left part is 96 − 8 = 88, the middle part is 48, and then take away 64 from 884,800. (6) 1,006,012,008: the parts are 1,006, then 012, then 008.
Cubes are built from multiplication facts, so keep those sharp with the multiplication game and the times tables chart. To see how squares work before tackling cubes, read squaring numbers fast. All the methods in this series are listed in our Vedic Maths guides.
If it is close to 10 or 100, use Yavadunam: the number plus twice the difference, then three times the difference squared, then the difference cubed. For any other two-digit number, Anurupyena's row of four numbers works every time.
The cube of a two-digit number contains three lots of each middle term. The first row gives one lot, and doubling adds the other two.
Treat the difference as negative. The left part goes down, the middle part stays positive because it uses the difference squared, and the cube of the difference is taken away at the end.