Mental Math with Tricks & Shortcuts

Dice Subtraction Math Trick

2012-03-09T07:46:00.000+06:00

This is agood math magic trick for kids who could use a little subtraction practice.It gives them an incentive to do subtraction because that is how the trickworks. Good for Kids in 2nd to 3rd grades but older students like it too.

Start with two dice. Have your subject roll the dice and then stack them one on top of the other. Now there are 3 numbers you cannot see:

the bottom of the top dice

the top of the bottom dice

the bottom of the bottom dice

Make it a point to let the person know that you cannot see those faces of the dice. Here is where the math magic comes in.

Tell them you'll be able to tell them how much the three hidden faces add up to. In your head subtract the very top face of the two dice from 14. This is the number that the other three will add up to. For example, if the top number is 6, the other three will add up to (14-6 = 8) 8! Pretty Cool!

You can do this with three dice also. Instead, just subtract the top number from 21.

Tips:

Every opposite 2 sides of a dice sums up 7.

Two interesting properties of prime numbers

2012-03-07T06:47:00.000+06:00

The Prime Number Trick

Let’shave a game.

Askyour friends to pick any prime number bigger than 5, but they must not tell youwhat it is.

Squareit. (In other words multiply the prime number by itself.)

Add17

Divideby 12

Withoutknowing which prime number your friends picked, you can still tell them: Therewill be a remainder of 6.

If you square ANY primenumber bigger than 5, then add 17 to it and then divide the result by 12, therewill always be a remainder of 6!

Forexample, if you want to try the trick with the prime number 37, here's what topush:

37* 37 = 1369 + 17 = 1386 / 12 = 115, with 6 remainder.

Ifyou want to try with the prime number 2801, here's what to push:

2801* 2801 = + 17 = / 12 =

...and that's theanswer! Now find a new prime number and try it.

The square of prime number & 24mystery!

If you square ANY primenumber bigger than 3, then subtract 1, the answer always divides by 24!

E.g. 11x11 = 121 then121 - 1 = 120 and yes 120 does divide by 24.

Multiplying Numbers that are Close to base 100 (Part-2)

2012-03-05T05:13:00.001+06:00

In Multiplying Numbers that are Close to base 100(part-1), we saw how to multiply numbers that are below or above base 100 injust one line. Here in part-2, we will see how to multiply numbers ‘Verticallyand Cross-wise’ where one number is above the base and the other is below.

For multiplying such numbers:

Subtract (from the number which is above the base 100) or add (with thenumber which is below the base 100) crosswise to find the first 2 or 3 digitsand append ‘00’ to the right.
Then subtract the product of both the differences from the base 100 fromthis result.

Supposeyou want to multiply 102 by 97

102 is 2 morethan 100 and 97 is 3 less than 100.

So, subtract (102-3) oradd (97+2) crosswise to get 99.

Append 2 zerosto 99 to get 9900.

Now, subtractthe product of both the differences that is 6 (2x3=6) from 9900.

You get 9900-6 =9894

So, 102x 97 = 9894

Another example,

107x 98 =?

107 is 7 morethan 100 and 98 is 2 below 100.

So, subtract (107-2) or add (98+7) crosswise to get 105.

Append 2 zerosto 105 to get 10500.

Now, subtractthe product of both the differences (7x2=) 14 from 10500.

You get 10500-14= 10486

Multiplying Numbers that are Close to base 100 (part-1)

2012-03-03T08:38:00.000+06:00

You must have known the fastestand easiest method of multiplying any 2-digit numbers together in just one line by now.

Here's the smarter way of multiplying numbers closeto base 100.

Findthe difference between both of the numbers and 100.
Multiplythese two values together and write it down. Make sure the answer takes up 2place values. For a single digit result add a ‘0’ beforeit and for a 3-digit result,carry the left digit.
Subtractthe difference found in step-1 of one of the numbers with the remaining number.You can subtract either way; you will always get the same answer. Write the result.

Suppose you want to multiply 88 by 98.

With ‘Vertically and Cross-wise’ you can give theanswer immediately.

Both 88 and 98 are close to 100.
88 is 12 below 100 and 98 is 2 below 100.

You can imagine the sum set out like this:

Multiplying vertically both the differences (12 and2) from 100 results in 24.

And, 86 comes from subtracting crosswise: 88 - 2 = 86 (or 98 - 12 = 86: you cansubtract either way, you will always get the same answer).

So, 88 x 98 =8624

This is so easy it is just mental arithmetic.

Multiplication can also be carried out startingfrom the left, which can be better because we write and pronounce numbers fromleft to right. In this case, remember to carry, if any.

Example, 96 by 92

96 is 4 below the base and 92 is 8 below.

We can cross-subtract either way: 96-8=88 or92-4=88. This is the first part of the answer and multiplying the"differences" vertically 4x8=32 gives the second part of the answer.

Multiplyingnumbers just over 100

This works equally well for numbers above the base. Here we add thedifferences.

103 x 104 =?
The answer is in two parts: 107 and 12,
107 is just 103 + 4 (or 104 + 3), and 12 is just 3 x 4.
So, 103 x 104 = 10,712
Similarly 107 x 106 = 11,342

107 + 6 = 113 and 7 x 6 = 42

And, 105 x 111 = 11,655

For 205x211; just double the first part of the answer, because 200 is 2x100.

Hence, 205x211=43,255

Two interesting properties of Numbers

2012-03-01T09:17:00.003+06:00

Square of Sum = Sum of Cubes

Aninteresting property of numbers is that, the square of sum of first ‘n’ naturalnumbers is equal to the sum of their cubes:

(1 + 2 + 3+...+ n)² = 1³+ 2³+ 3³+...+ n³

Examples:

(1 + 2 + 3 + 4 + 5)²             = 225 =    1³ + 2³ + 3³ + 4³+ 5³

(1 + 2 + 3 + 4 + 5 + 6)²       = 441 =    1³+ 2³ + 3³ + 4³ + 5³ + 6³

(1 + 2 + 3 + 4 + 5 + 6 + 7)² = 784 =   1³+ 2³ + 3³ + 4³ + 5³ +6³ + 7³

Sum of all the digitsof a Cube

The sum of any cube when reduced to a single digit will always beeither 1, 8, or 9.

For Example:

5³ = 125 = 1+2+5 =8
9³ = 729 = 7+2+9 = 18=1+8=9
17³ = 4913=4+9+1+3=17=1+7=8
4³=64=6+4=10=1+0=1

Also, they follow a cycle. The numbers 1 ,8, and 9 appear in order:

1 - sum of the digits of cube of 1 would be 1
2 - would be 8
3 - would be 9
4 - would be 1 (4³ = 64 = 6+4 = 10= 1+0 = 1)
5 - would be 8 (5³ = 125 = 1+2+5 = 8)
6 - would be 9 (6³ = 216 = 2+1+6 = 9)

So on and so forth.

Squaring numbers made up of repeating digit (part–4)

2012-02-28T09:13:00.000+06:00

Squaringnumbers made up of repeating '9'

In the previous 3part, we saw how to square numbersmade up of repeating 1, 3 and 6 respectively without doing any calculation. In thispart, we will see how to square numbersmade up of repeating 9.

1) Write one fewer 9 than the number of repeating 9's

2) After that put 8

3) Writ samenumber of 0's as 9's

4) Finallyplace 1

Let’s illustrate with an Example:

If thenumber to be squared is 999

1) There are 3 repeating 9's. So, put one fewer i.e. two 9’s-99

2) Write 8 - 998

3) Put two 0’s as9's - 99800

4) Finallyplace 1 - 998001

So, Square of 999 = 998001

See the pattern?

So, 99999²=?

1) There are 5 repeating 9's. So, write four 9’s - 9999

2) Put 8 - 99998

3) Write four 0’s as9's - 999980000

4) Finallyplace 1 - 9999800001

So 99999² = 9999800001

Squaring numbers made up of repeating digit (part–3)

2012-02-26T09:23:00.000+06:00

Squaring numbers made up of repeating'6'

Inpart-1 and part-2 of squaring numbers made up of repeating digit, I discussed how to square numbersmade up of repeating 1 and to square numbers made up of repeating 3 respectively. In the part-3,we will see the techniqueto square numbers made up of repeating 6’s (sixes).

1) Writeone fewer 4 than the number of repeating 6's

2) Afterthat place 3

3) Putsame number of 5's as 4's

4) Finallyplace 6

Let’sillustrate with an Example:

If the number to be squared is 666

1) Thereare 3 repeating 6's. So, write one fewer i.e. two 4’s - 44

2) Put3 - 443

3) Writetwo 5’s as 4's - 44355

4) Finallyput 6 - 443556

So, Square of 666 = 443556

See thepattern?

Square of 66666 =?

1) Thereare 5 repeating 6's. So, put four 4’s - 4444

2) Place3 - 44443

3) Putfour 5’s as 4's - 444435555

4) Finallyplace 6 - 4444355556

So 66666² = 4444355556

Is 0.999… really equal to 1 ?

2012-02-24T08:47:00.000+06:00

Themeaning of 0.999… is a tricky concept, and depends on what we allow a number tobe.

Formost mathematicians, 0.999… represents a series of numbers: 0.9, 0.99, 0.999,0.9999, and so on

Acommon assumption is that numbers cannot be “infinitely close” together— they’re either the same, or they’re not.

Ifwe assume infinitely small numbers don’t exist, we can show 0.999… = 1.

Fornon-math persons, you will probably disagree with the equality, but there aremany elementary proofs that could show it.

Proof1:

Multiplying both sides by 9 we have

Proof2:

But ,therefore,

Proof3: This is the mostintuitive proof.

              Let k =0.999...
              Then 10k = 9.999...
              10k - k = 9.999... - 0.999...
              9k = 9
              k = 1

And k = .999...
Hence, 0.999... = 1

Squaring numbers made up of repeating digit (part – 2)

2012-02-22T09:48:00.000+06:00

Squaring numbers made up of repeating ‘3’

Inpart-1 of squaring numbers made up of repeating digit, I discussed the simplest method of squaring numbersmade up of repeating 1. Here in part-2, we will see the technique to squarenumbers made up of repeating 3’s (three) without doing any calculation.

Write one fewer '1' than the number of repeating 3's
After that put 0
Write same number of 8's as 1's
Finally place 9

Let’s illustrate with an Example:

If thenumber to be squared is 333

There are 3 repeating 3's. So, put one fewer that is two 1’s - 11
Place 0 - 110
Write two 8’s as1's - 11088
Finallyput 9 - 110889

So, 333²= 110889

See the pattern?

So, Square of 33333 =?

There are 5 repeating 3's. So, write four 1’s - 1111
Place 0 - 11110
Put four 8’s -111108888
Put9 - 1111088889

So, 33333² = 1111088889

Squaring numbers made up of repeating digit (part–1)

2012-02-20T11:07:00.000+06:00

Squaring numbers made up of repeating '1'

Hereis a cool trick to square numbers made up of repeating 1 (one).

Count the number of repeating 1 in the number to be squared and put the digits ascending from 1 up to that number. That is, if there are four 1's in the number to be squared, put 1234.
Then, next to the highest digit, place the same digits inversely, that is, 321.

That’sit. It is so easy, it’s just mental math. Easy and fast.

Let’sillustrate with an Example:

If the number to besquared is 111

There are 3 repeating 1's. So, put 123.
Then, place 21 after 123.

So, 111² = 12321

Seethe pattern?

Let's see another one,Square of 11111 =?

There are 5 repeating 1's. So, write 12345.
Then place 4321

So, 11111² = 123454321

More example?

1² = 1

11² = 121

111² = 12321

1111² = 1234321

11111² = 123454321

111111² = 12345654321

1111111² = 1234567654321

11111111² = 123456787654321

111111111² = 12345678987654321

Faster way to multiply any number with numbers from 12-19

2012-02-18T10:32:00.002+06:00

You likely know the easiest method to multiply any number by 11 by now. Here is the extension form of the method for Multiplying any number with numbers from 12-19 in the faster way.

The basic recipe for multiplying anynumber by 12-19 is to “multiply one’s digit of the number from 12-19 withall the digits of the other number one by one and add to its neighbor on theright.”

Let’s multiply 346 by 17.

Start with the right-most digit, 6. Multiply 7 with 6 and add the neighbor(none in this case, as 6 is the right-most digit). We get 42.
Write down 2,and carry the 4. -----> 2

Move left to the next digit, 4. Multiply 7 with 4 to get 28, add the neighborto right 6, to get 34, and add the carry 4, to get 38.
Write down the 8,and carry the 3. -----> 82

Move left to the next digit, 3. Multiply 7 with 3 to get 21. Add the neighbor,4, and we get 25. Add the carry 3, to get 28.
Write down the 8, andcarry the 2. -----> 882

Now add the carry 2 to the left most digit 3 which gives us 5.
Finally, Writedown 5. -----> 5882

So, our answer is 5882.

Let’s try another example: 123456×13.

First digit is 6. Multiply 3 with 6 + no neighbor --> (3x6+0) = 18. Write 8 and carry 1. ---> 8

Nextdigit is 5. Multiply 3 with 5 + neighbor 6 + carry 1 (3x5+6+1) = 22. Write 2 and carry 2. ---> 28

Nextdigit is 4. Multiply 3 with 4 + neighbor 5 + carry 2 (3x4+5+2) = 19. Write 9 and carry 1. ---> 828

Nextdigit is 3. Multiply 3 with 3 + neighbor 4 + carry 1 (3x3+4+1) = 14. Write 4 and carry 1. ---> 4828

Nextdigit is 2. Multiply 3 with 2 + neighbor 3 + carry 1 (3x2+3+1) = 10. Write 0 and carry 1. ---> 04828

Nextdigit is 1. Multiply 3 with 1 + neighbor 2 + carry 1 (3x1+2+1) = 6.
Write 6 and no carry. ---> 604828

Finally,the left-most digit 1 + no carry = 1. Write down 1. ---> 1604828

The answer is 1604828.

Onefinal example: 35246×15

5x 6 = 30. Write down 0, carry 3. -------------> 0

5x 4 + 6 + 3 = 29. Write down 9, carry 2. ---> 90

5x 2 + 4 + 2 = 16. Write down 6, carry 1. ---> 690

5x 5 + 2 + 1 = 28. Write down 8, carry 2. ---> 8690

5x 3 + 5 + 2 = 22. Write down 2, carry 2. ---> 28690

3+ 2 = 5. Write down 5. -------------------- > 528690

So, 35246×15= 528690

Digging out anybody’s age

2012-02-16T11:04:00.000+06:00

The Amazing Age Revelation Trick

Firstof all, pick the number of days a week that you would like to eat out.

Multiplythis number by 2.

Add 5.

Multiplyit by 50.

If youhave already had your birthday this year . . . add 1762

If youhave NOT had your birthday yet this year . . . add 1761

Subtractthe four digit year that you were born.

Youshould now have a three digit number:

Thefirst digit of this was your original number (i.e. how many times you want togo out to eat each week).

Thesecond two digits are your age!!!

Youcan also amaze your friends or family members by asking them to pick a numberbetween 1 and 10 and then follow the whole process.

2012 is the only year it will ever work, so spread the fun while it lasts...
However, for perpetual...

Add 1763 instead of 1762 for 2013, 1764 for 2014 and so on…if youhave already had your birthday that year

And, if you have NOT had your birthday yet that year, add 1762 instead of 1761for 2013 and 1763 for 2014 and so on…

An amazing multiplication number puzzle

2012-02-14T10:41:00.000+06:00

If you multiply two five digit numbers you can getthe answer 123456789. Can you guess the two five digit numbers?

Thisis a very beautiful puzzle and you might think that a big mathematical theoryshould be hidden behind it. But in fact its beauty is only incidental.

Give it a try ...
        .
        .
        .
Hereis the Hint to help you solve the puzzle,

      'Decomposethe number (123456789) into its prime factors'.

            123456789= 3 x 3 x 3607 x 3803

Got it?

OK, check it with the solution … scroll down
         .
         .
         .
         .
Theanswer is:

123456789= (3 x 3607) x (3 x 3803) = 10821 x 11409

Finding the number of factors of any number

2012-02-12T15:23:00.000+06:00

Ifa number can be expressed as a product of two whole numbers, then the wholenumbers are called factors ofthat number.

Inother words, A factor is a whole number which divides exactlyinto a whole number, leaving no remainder.

Forexample, 13 is a factor of 52 because 13 divides exactly into 52(52 ÷ 13 = 4 leaving no remainder).

52 = 1 x 52 = 2 x 26= 4 x 13

So,the complete list of factors of 52 is: 1, 2, 4, 13, 26, and 52 (all thesedivide exactly into 52).

A simple technique to find the number of factors of a givennumber is to express the number as a product of powers of primenumbers or prime factors.

To illustrate let’s find the numbers of factors of ourexample 52.

Note that, 52 can be expressed as 4 x 13 = 2²x 13

So, the prime factors of 52 are 2 and 13.

Now, increment the power of each of the primenumbers by 1 and multiply the result.

In this case it will be (2 + 1) x (1 + 1) = 3 x 2 =6 (power of 2 is 2 and power of 13 is 1)

Therefore, there will 6 factors including 1 and 52.

Alsonote that, all numbers have a factor of 1 since 1 multiplied by any numberequals that number. All numbers can be divided by themselves to produce thenumber 1. Therefore, we normally ignore 1 and the number itself as usefulfactors.

So, excluding, these two numbers, you will have (6– 2) = 4 factors.

To be certain the factors are: 2, 4, 13 and 26.

To further illustrate let’s find the numbers of factorsof 48.

48 can be written as 16 x 3 = 2⁴ x 3

So, the prime factors of 48 are 2 and 3.

Now, increment the power of each of the primenumbers by 1 and multiply the result.

In this case it will be (4 + 1) x (1 + 1) = 5 x 2 =10 (the power of 2 is 4 and the power of 3 is 1)

Therefore, there will be 10 factors including 1 and48.

Excluding, these two numbers, you will have (10 – 2)= 8 factors.

And the factors are: 2, 3, 4, 6, 8, 12, 16 and 24

Another example:

216 = 8 x 27 = 2³ x 3³

Here also, the prime factors of 216 are 2 and 3.

Now, increment the power of each of the primenumbers by 1 and multiply the result.

(3+1)x (3+1) = 4 x 4 = 16(power of both 2 and 3 is 3)

Therefore, there will be 16 factors including 1 and216.

Excluding, these two numbers, you will have 16 – 2= 14 factors which are 2, 3, 4, 6, 8,9, 12, 18, 24, 27, 36, 54, 72 and 108.

Multiplying 2 numbers together that differ by a small even number

2012-02-10T15:22:00.000+06:00

If you’re good at memorizing some squares and use the tricks for some kinds of numbers,you’ll be able to quickly multiply together many pairs of numbers that differby 2, or 4, or 6.

Let’s say you want tocalculate 12×14.

When two numbers differ by2 their product is always the square of the number in between them minus 1.

12×14 = (13×13)-1 = 169-1 = 168.

16×18 = (17×17)-1 = 289-1 = 288.

99×101 = (100×100)-1 =10000-1 = 9999

If two numbers differ by 4then their product is the square of the number in the middle (the average ofthe two numbers) minus 4.

11×15 = (13×13)-4 = 169-4 =165.

13×17 = (15×15)-4 = 225-4 =221.

If the two numbers differby 6 then their product is the square of their average minus 9.

12×18 = (15×15)-9 = 225-9 = 216.

17×23 = (20×20)-9 = 400-9 = 391.

This trick works effectively, ifyou’ve memorized or can quickly calculate the squares of numbers.

The Missing Digit Trick !

2012-02-08T18:59:00.000+06:00

Ask a friend to secretly write down ANY number (at least fourdigits long). e.g. 78341

Ask the friend to add up the digits... e.g. 7+8+3+4+1 = 23

... and then subtract the answer from the first number. e.g. 78341- 23 = 78318

Your friend then crosses out ONE digit from the answer. (It can beany digit except a zero) e.g. 7x318

Your friend then reads out what digits are left .e.g. 7-3-1-8

Even though you haven't seen any numbers, you can say what themissing digit is! EIGHT

THE SECRET

This great trick relies on the power of 9. How?

After your friend has added up the digits and subtracted them, theanswer will ALWAYS divide by 9. If a number divides by nine, then when you addthe digits up, they will also divide by 9. If you check our example 7+8+3+1+8 =27 which does divide by nine.

When your friend crosses a digit out, he then reads out the digitsthat are left. You add them up. In the example we had 7+3+1+8 = 19

All you do now is see what you have to add on to your answer toget the next number that divides by nine! The next number to divide by 9 after19 is 27. So you need to add on EIGHT. This is the number that was crossed out!

Here's another example:

Say the number written down is 873946284 (urk!).
Your friend adds the digits 8+7+3+9+4+6+2+8+4 = 51
Your friend does the subtraction: 873946284 - 51 = 873946233
(So far you have NO IDEA what numbers are whizzing around!)
Your friend crosses a digit out 87394x233 and tells you what's left.
You add 8+7+3+9+4+2+3+3 = 39.

The next number that divides by 9 after 39 is 45. As 45-39=6, this means that6 is the missing digit!

The 4-2-1 Loop

2012-02-06T17:22:00.001+06:00

A Math-magical Loop

Pick a whole number and enter it into your calculator.

If it is even, divide by 2. If it is odd, multiply by 3 and add 1.
Repeat the process with the new number over and over. What happens?
The sequence always ends in the "loop": 4.....2.....1.....4.....2.....1...

Let’sillustrate with an Example:

Start with 17

17is odd, so we multiply by 3 and add 1. We get 52. (17x3=51+1=52)
52is even, so we divide by 2. We get 26. (52 / 2 = 26)
26is also even, so we divide by 2 again and get 13.
13is odd, so we multiply by 3 and add 1. We get 40. (13x3=39+1=40)
40is even, so we divide by 2. We get 20. (40 / 2 = 20)
20is even, so we divide by 2 and get 10.
10is also even so we divide by 2 again and get 5.
5is odd so we multiply by 3 and add 1. We get 16.
16is even, so we divide by 2 and get 8.
8is also even so we divide by 2 again and get 4.
4is even so we divide by 2. We get 2.
2is even, so we divide by 1 and get 1.
1is odd, so we multiply by 3 and add 1. We get 4.
4is even so we divide by 2. We get 2. And so we begin the loop
4.....2.....1.....4.....2.....1...

The Secret of 73 and 137 !

2012-02-04T16:15:00.000+06:00

TheSecret of 73

Let’s havea game.

For this trick, secretly write 73 on a piece of paper, fold it up, and give to an unsuspecting friend.

Now have your friend select a four-digit number whatever (say, 3125) and enter it twice into a calculator. (31253125)
Announce that the number is divisible by 137 and have him verify it on his calculator.
Next, announce that he can now divide by his original four-digit number. After he has done so, dramatically command him to look at your prediction on the paper. It will match his calculator display: 73!

Want to know the simplemath behind this "TRICK"?

Entering a four-digit number twice (31253125) is equivalent tomultiplying it by 10001. (3125 x 10001 = 31253125). Since 10001 = 73 x 137, theeight-digit number will be divisible by 73, 137, and the original four-digitnumber.

You can do the process otherway round to ploy 137 as the secret.

Squaring any number close to base 100

2012-02-02T12:52:00.002+06:00

You likelyknow the easiest method of squaring any 2-digit numbersby now. Here is a technique to find square of any number from 76-125 that is nearthe base 100.

To square any number near base 100 just follow 3 steps as below:

1) Find the surplusor deficit from base 100
2) Add the surplus(if it is more than base) with or subtract the deficit (if it is less than base)from the whole number given and put the result.
3) Now, sincebase 100 has 2 zeros so, 2 more digits to go. Find the square of surplus ordeficit. Put the result in the last 2 places (and add the 3^rd left-digit,if any to the number derived from step-2).

That’s it! Simple and fast.

Let us do some examples to illustrate the technique:

What is the square of 104? Or, 104²= ?

To Find the square of 104 thefaster way all u need to do is:
1) Find the surplusor deficit from base 100. It is 4 surplus (104–100=4).
2) Now,since 104 is 4 more than base 100 so, add 4 to the entire number, which is104+4 = 108. Put 108 as the first 3 digits of the answer.
3) Now, 2more digits to go. Find the square of surplus 4 which is 16. Put 16 asthe last 2 digits of the answer.

So, our final answer is 10816

Now what if the number that you are trying to square is below the basethat is below 100?

Once again, the same thing. Let’s illustrate with an example:

Find 97²

1) Find the surplusor deficit from base 100. It is 3 deficit (100 – 97 = 3).
2) Now,since 97 is 3 less than base 100 so, deduct 3 from the entire number, which is 97-3= 94. Put 94as the first 3 digits of the answer.
3) Now, 2more digits to go. Find the square of deficit 3 which is 9. Put 09 asthe last 2 digits of the answer. Notice the extra ‘0’. An extra 0 is to beadded to make it a 2 digit number.

So, our final answer is 9409

Further example: 107²= ?

Add the distance (7) from the base 100 to the entire number.

7 + 107 = 114.

Now, 2 more digits to go. Square of 7 is 49.

So, the answer is 11449.

113²= ?

Once again add surplus 13 that is distance from the base 100 to theentire number i.e. 113.

So, 113+13 = 126. These form thefirst 3 digits of your answer.

Now for the last 2 digit you need to find the square of the distancethat is 13.

Square of 13 is 169.

Now put 69and add 1 to 126 as we have left 2 places to go to complete the square.

1+126 = 127

The answer is 12769.

89²= ?

89 is 11 less than the base 100.

subtract 11 from 89. It would be 78,which are the first 2 digits of the answer.

Now, put the square of the deficit 11, which is 121.

Put 21 in the last 2 places and add 1 to 78to make 79

So the answer would be 7921.

Try 88²

100 - 88 = 12
88 - 12 = 76. 76__.
12² = 144. What now? Just carry the 1 over the 6.

76__

_144
7744

Find 112²

112 - 100 = 12
112 + 12 = 124. 124__.
12² = 144. Carry the 1 over the 4.

124__

__144
12544

The Mysterious 495 ! - ‘Kaprekar’s Constant’

2012-01-30T23:52:00.000+06:00

Let’shave an interesting mathematical curiosity.

Think of a 3-digit number where not all the digits are the same.
Arrange the digits in descending order and ascending order from left to right. This will give you two 3-digit numbers — one the smallest number that you form from the digits, and the other the largest.
Subtract the smaller number from the larger number.
Forthe resulting number, repeat the procedure (step 2) and keep repeating.

Ifyou repeat this process over and over again, you will end up with 495!

Youdon’t believe it? Let’s have some examples:

Let's take 729 to work through. The routine says:
1. Make the largest number (972) and the smallest number (279)
2. Subtract the smaller number from the larger (972 – 279 = 693)
3. Repeat the procedure for 693, and keep repeating.

What happens is the following:

In the example, notice we end up with the number 495, and that number justgoes to itself. If the difference contains the digits 9, 5, and 4, you will end up with 495 ineach repetition.

But the more amazing thing is this: every three digit number whosedigits are not all the same will eventually hit 495!

In fact, it even happens for numbers like 7 if you add on leading zeros (so7 is really 007, the numbers you can make are 700 and 007. Subtracting leads to693, which we showed above ultimately results in 495).

The only numbers that do not work are ones where all the digits are the samelike 111, 222, and so on, as they obviously result in the number 0.

Example 2

Our number is 592.
Arranging it in descending and ascending number, we have 952 and 259.
Subtracting 259 from 592 we get 693.
Repeating steps 2-3, we have 963 -369 = 594
Repeating steps 2-3 again, we have 954 – 459 = 495
Repeating steps 2-3 again, we have 954 – 459 = 495

Example3

Our number is 466
Arranging in ascending and descending we have 664 and 466.
Subtracting 466 from 664 we have 198.
Repeating steps 2-3, we have 981 – 189 = 792.
Repeating steps 2-3, we have 972 – 279 = 693. And this is the same as the number 3 in Example 1 above, which will end up in 495.

Example4

Let’s have 990.
Arranging the digits in ascending and descending, we have 990 and 099 or simply 99.
Subtracting, we have 990 – 99 = 891.
Repeating steps 2-3, we have 981 – 189 =792. And this is the same as difference in Example 2 step 4. This means that it will also end up in 495!

The number 495 is the‘Kaprekar’s Constant’ for three digit numbers. Yes, if you choose any 3-digitnumber, you will end up in 495 (try other numbers). The more famous ‘Kaprekar’sConstant’ is 6174 for 4-digit numbers.

Converting Repeating Decimals Into Fractions

2012-01-28T12:24:00.002+06:00

It is fairlyeasy to take a fraction and turn it into a repeating decimal by hand. This issimple division.

But being able to go in the other direction and turn arepeating decimal into a fraction would be a cool party trick.

Let us take "0.454545454545..."to illustrate the trick.

Simply take the number that repeats, in this case 45, and divide it by howevermany 9's as there are digits in the original number. Since 45 is 2 digits, then divide 45 by 99.

Now, 45/99 simplifies down to 5/11.

So, "0.454545454545..." = 5/11

For more understanding, if the repeating number is 0.4564564564..., then divide456 by 999 since it has 3 digits (which would simplify down to 152/333.

Simple!

Just makesure if you perform this as a "magic trick" of sorts, to simplifyyour fraction down before you blurt it out, to conceal the magic.

Whydoes this work?

0.123123123..... = 41/333

Let, X = 0.123123123.....

So, 1000X =1000 * 0.123123123..... = 123.123123123....

Now, 1000X- X = 123.123123123.... - 0.123123123.....

> 999X =123.000

Therefore, X= 123/999 = 41/333

Adding Time Trick

2012-01-26T17:23:00.002+06:00

A cool trick to add hours and minutes together!

Here is a cool trick to add hours and minutes together:

Let's try adding 3 hr 23 minutes and 2 hr 46 minutes together.
Make the 3 hr 23 minutes into one number, which will give us 323 and do
the same for the other number, 2 hours 46 minutes, giving us 246.

Now add these two numbers together:

              323
              246
               569

So, we have a total of 569.

Now,just add the 40 time constant to the total, no matter what the hours andminutes are.

569 + 40 = 609

And now, split the number to get the answer which is 6 hrs and 09 minutes!

P.S. Add 40 to the total in case the minutes are equal to or greater than 60, otherwise keep it same.

Multiply Up to 20 X 20 in Head

2012-01-25T11:09:00.001+06:00

Multiplyingtwo 2-digit numbers between 11 and 19

With this trick, you will be able to multiplyany two numbers from 11 to 19 in your head quickly, without the use of acalculator.

Here arethe Rules:

1)Prefix the number to be multiplied by zero.

2)Thefigures in the answer is obtained by adding the units place of number (11-19)with the each figure of the number to be multiplied and adding the successivenumber to it. (This should be done similar to our multiplication from right toleft)

Try this:
Take 15 x 13 for an example.

Always place the larger number of the two on top in your mind.
Then draw a shape mentally that covers the 15 and the 3 from the 13 below. Those covered numbers are all you need.
First add 15 + 3 = 18
Add a zero behind it (multiply by 10) to get 180.
Multiply the covered lower 3 x the single digit above it the "5" (3x5= 15)
Add 180 + 15 = 195.

Let’s try another example:

17 x 18 = ?

Step 1: 18 + 7 = 25

Step 2: Add a zero behind it (multiply by 10) to get 250.

Step 3: Multiply the covered lower 7 x the single digit above it the "8" (7x8= 56)

Step 4: Add 250 + 56 = 306

So, 17 x 18 = 306

Simple!

Multiplying any 2 or 3-digit number by 111

2012-01-21T00:24:00.001+06:00

You likely know the techniqueof multiplying any number by 11. Here is the trick to multiply a 2-digit and 3-digitnumber by 111.

Multiplying a 2-digit number by 111

Example: 111 × 58

1. Writedown the ones digit. > 8

2. Add thetens and ones; carry, if any. > 38 (5 + 8 = 13)

3. Repeatthe previous step (and add carry, if any) > 438 (5 + 8 + 1 = 14)

4. Write thetens digit (add carry, if any) > 6438 (5 + 1 = 6)

So, 111 × 58 = 6,438

Multiplying a 3-digit number by 111

Useright-to-left “sweeps” of 1, 2, 3, 2, and 1 digit.

Example:111 × 234

1. Writedown the right most digit. > 4

2. Add the2 right most digits (carry, if any) > 74 (3 + 4 = 7)

3. Add all the 3 digits (and carry, if any) > 974 (2 + 3 + 4 = 9)

4. Add the2 left most digits (and carry, if any) > 5974 (2 + 3 = 5)

5. Writedown the left most digit (add carry) > 25974

So, 111 × 234 = 25,974

Multiplying any 3-digit numbers between 100 and 109

2012-01-19T12:41:00.000+06:00

Multiplying any 3-digit numbers between 100 and 109 together is a 3-step simple technique:

Step 1: Put 1 as the first digitof the answer. It will always be 1.

Step 2: Add the last digits ofeach numbers. That’s the next two digits of the answer. Ifthe addition results in a single digit number, make sure to put an extra zerobefore it.

Step 3: Multiply the last digitsof each number. That’s the last two digits of the answer. Here also make sureto put an extra zero if the product is a single digit number.

Example: 107 x 105 = ?

107

X 105

The first number will always be 1
107
x 105
1#### (1x1 = 1)
Add the last digits of each numbers. 7+5=12. That’s the next two digits of the answer.
107
x 105
112## (5+7 = 12)
Lastly, multiply the last digits of each number, 7x5=35
    107
x   105
11,235   (5x7 = 35)

The answer is11,235.

So,107 x 105 = 11,235

Another Example:

108

X 109

11,772

Fibonacci Number Trick

2012-01-18T11:56:00.001+06:00

The Fibonacci numbers are a series of whole numbers in which each number is the sum of the two preceding numbers. Beginning with 0 and 1, the sequence of Fibonacci numbers would be 0,1,1, 2, 3, 5, 8, 13, 21 etc.

You can easily amaze yourfriends and family members by being able to tell them the sum of any 10 Fibonacci numbers orally at the blink of eye.

First, tell them to take aFibonacci sequence of 10 numbers (as above) behind you. This needs them to write down the numbers sequentially. Now, tell them to let you look the sequence for once and you notice the 7th number and you can easily tell them the sumof those 10 numbers mentally simply by multiplying the 7^th number by 11 mentally. Andyou know the technique to multiply any number by 11 mentally. So, it'sjust a matter of blink of eyes!

Here is an example to illustrate the technique:

Step 1: Choose any two numbers

Step 2: Form a Fibonacci sequence for ten numbers

Let us choose 5 for the first number and 6for the second number.

Then add the numbers to get a Fibonacci sequence. 5+6 gives the3rd number which is 11; 6+11 gives the 4th number which is 17. Theentire sequence is as follows:

1st - 5
2nd - 6
3rd - 11
4th - 17
5th - 28
6th - 45
7th - 73
8th - 118
9th - 191
10th - 309

The trick is to ...

multiply the 7th number by 11 (youalways multiply it by 11). The answer you get when you multiply the 7thnumberby 11 will always be the same result as if you had added all ten numbers. Inthe above example, the 7th number is 73. Multiply 73 by 11 and the answeris 803. If you add all 10 numbers together the answer will also be 803.

Try this with your own numbers.

Interesting Propertyof the Fibonacci sequence

In the Fibonacci sequence, if you take any three consecutivenumbers, add them, and divide the sum by 2, you always get the third number.

Let me explain:

1+2+3=6 and 6/2=3
5+8+13=26 and 26/2=13
21+34+55=110 and 110/2=55
89+144+233=466 and 466/2=233
377+610+987=1974 and 1974/2=987

and I assume it goes on to infinity.

Multiplying any 2 digit numbers together

2012-01-15T20:43:00.000+06:00

Multiplying any 2 digit numbers together by mere mental arithmetic!

GTFT5TF7CZDT

An easy way to multiply a 2 digit number by a 2 digit number using RainbowMultiplication.

Step1: Multiply the ones digit together. This gives the first figure of the answer.

Step2: Cross-multiply and add. This gives the middle figure.

Step3: Multiply the tens digit together. This gives the last figure of the answer.

Example: 12 x 23

Sothe answer is 276
This is normally
called longmultiplication but actually the answer can be written straight down using the VERTICALLY AND CROSSWISE formula.

21 x 23 = 483

Wefirst put, or imagine, 23 below 21:

Thereare 3 steps:

a)Multiply vertically on the left: 2 x 2 = 4. This gives the firstfigure of the answer.

b) Multiply crosswise and add: 2 x 3 + 1 x 2 = 8. This gives themiddle figure.

c) Multiply vertically on the right: 1 x 3 = 3. This gives thelast figure of the answer.

Similarly 61 x 31 = 1891

6 x 3 = 18; 6 x 1 + 1 x 3 = 9; 1 x 1 = 1

Carries

There were no carries inthe method given above. However, there only involve one small extra step, ifthere is any carries in any place.

21 x 26 = 546

The method is the same asabove except that we get a 2-figure number, 14, in the
middle step, so the 1 is carried over to the left (4 becomes 5).

So 21 × 26 = 546

33 x 44 = 1452

Theremay be more than one carry in a sum:

Verticallyon the left we get 12. Crosswise gives us 24, so we carry 2 to the left
and mentally get 144.

Thenvertically on the right we get 12 and the 1 here is carried over to the 144 tomake 1452.

Dividing any number By 9, 90, 900 and so on

2012-01-04T16:33:00.000+06:00

Dividing any number by 9

Thetechnique for Dividing any number by 9 mentally is simply to reduce a complex divisionto a very simple addition. The technique can be applied from both ends i.e. from right-most digit or from left-most digit.

Dividingby 9 into a mixed number from right-most digit uses the Divisibility Rules for9:

1. First, add all the digitstogether and divide by 9, keeping in mind the whole number and the remainder.

2. Write the remainder over 9, thisis the fraction part of the answer. (Make sure the fraction is in simplestform.)

3. Add all the digits togetheragain, except for the last one. If there was a whole number from step 1, addthis too. Write the number down, carrying if need be.

4. Continue to add all the digitstogether, subtracting off one digit each time, until you are left with thefirst (left most) digit. Write these numbers down, carry if need be.

5. Write the last number down.(Remember to add any carried numbers.)

Here are some examples to helpillustrate the above steps:

1234 ÷ 9 = __________(mixed number)

a. Addall the digits together and divide by 9: 1+2+3+4 = 10. 10 divided by 9 is 1with 1 remainder.

b. Write1/9, this is the fraction of the answer.

c. Add1+2+3+1 = 7. Write 7.

d. Add1+2 = 3. Write 3.

e. Thelast number is 1, so write 1.

f. Theanswer is 137 1/9.

8346 ÷ 9 =__________(mixed number)

a. Addall the digits together and divide by 9: 8+3+4+6 = 21. 21 divided by 9

is 2 with3 remainder.

b. Thefraction part is 3/9, which reduces to 1/3.

c. Add8+3+4+2 = 17. Write 7, carry 1.

d. Add8+3+1 = 12. Write 2, carry 1.

e. 8+1 =9. Write 9.

f. Theanswer is 927 1/3.

For dividing by 90, move thedecimal point to 1 digit left and for dividing by 900, move the decimal pointto 2 digits left after diving by 9 or following the above steps.

However,the faster way to divide any number by 9 mentally is to start it from left-most digit. Here are some examples to helpillustrate the method:

221013 ÷ 9 = ?

Start putting the first (left) digit of the dividing as it is which is 2,and you have the first answer digit.

The next one would be this answer digit plus the next digit of thedividend. So, 2 plus 2 is 4.

The next answer digit would be this 4 plus 1 that is 5.

Now, using the same logic, the next answer digit would be 5 plus 0,which is again 5.

Next one would be 5 plus 1 that is 6.

And then the last one would be 6 plus 3 that is 9. But do not put 9 straightaway,because this would be the remainder, if there is a remainder. In this case,there would be none, because 9 still goes 1 times in 9. This 1 would be addedto the last findings i.e. 6 resulting 7.

So, the answer would be 2 > 4 > 5 > 5 > (6+1) =7 that is 24557.

32142 ÷ 9= ?

The answer would be 3,

3 plus 2 that is 5,

5 plus 1 that is 6,

6 plus 4 that is 10, now 10 is a 2 digit number so carry forward 1 (tobe added with 6 making 7) and put 0 here.

And 10 plus the last digit 2 would be 12, so do notput 12 straightaway. Now, 9 goes how many times in 12? Nine 1 times is 9. So, this 1would be added to 0 and 3 is the remainder.

So, the answer would be 3 > 5> (6+1) =7 > 1 that is 3571 with 3reminder.

This way you can reduce a complexdivision to a very simple addition. And everyone is really very good at addingnumbers.

Dividing any number by 5, 50, 500 and so on

2011-12-29T17:54:00.002+06:00

Dividing a large number byfive is actually very simple. All you do is multiply by 2 and move the decimalpoint to 1 digit left. To divideby 50, move the decimal point to 2 digits left and to divide by 500, move thedecimal point to 3 digits left after multiplying the number by 2. Example:

a) 195 ÷ 5

Step1:195 × 2 = 390
Step2: Move the decimal: 39.0 or just 39

b) 2978 ÷ 5

Step1: 2978 × 2 = 5956
Step2: 595.6

To divideby 50, move the decimal point to 2 digits left and to divide by 500, move thedecimal point to 3 digits left after multiplying the number by 2.

45289 ÷ 5 =9057.8

45289 ÷ 50 = 905.78

45289 ÷ 500 = 90.578

Extracting Square Roots

2011-12-20T11:01:00.001+06:00

Faster way to extract the square root of any perfect number

Finding the square rootof a number is the inverse operation of squaring that number. Remember, thesquare of a number is that number times itself.
Square of n = n²
Square of 5 = 5²= 5 x 5 = 25
Therefore, the square root of 25 is 5

The perfect squares are the squares of the wholenumbers such as 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 so on and so fort.

The faster way to extract the square root of any perfect number is illustrated here.

Before learning theprocedure, it is wise that the performer memorizes the squares of thenumbers 1-10 which is very elementary:

1² = 1

2² = 4

3² = 9

4² = 16

5² = 25

6² = 36

7² = 49

8² = 64

9² = 81

10² = 100

Technique– 1

To extract the square root of any perfect square, followthe below steps:

Step 1: Omitthe digit next the last digit.

Step 2: Find the square root of its extremities. Todo this, always start with right side. (If the right digit is not a perfectsquare, then start from the left extremity) Find the square root of that and writeit down.

Step 3: Subtractthis finding from 10. Write the result down too.

Step 4: Find the square root of left extremities.This is the tens digit of final answer.

Step 5: Write the result 2 times. Once with thesquare root of right digit and secondly with the number found after subtractingthat from 10. Now, there are 2 possible answers one greater and one smaller; bothwith the same tens digit.

Step 6: To find the right answer, multiply thesquare root of left extremities with its successor. If left extremities aregreater than the product, the right answer would be the greater option and if leftextremities are less than the product, the right answer would be the smaller option.It’s so simple!

Let us illustratethe method with some examples:

Extractingsquare root of 441 (√441)

1) Skip the digit next the last digit which is 4 (√4x1)

2) Square root of right extremity i.e. 1 is 1

3) 10 minus 1 is 9

4) Square root of left extremity i.e. 4 is 2

5) Write down 2 as tens digit with both 1 and 9 to get2 possible answers (21 and 29)

6) Now, 2 times its successor which is 3 (2x3) is 6. Theleft extremity which is 4 is less than 6. Therefore, the right answer must be thesmaller option which is 21.

Let’s go toanother example: √784 (square root of 784)

1) Skip the digit next the last digit which is 8 (√7x4)

2) Square root of right extremity which is 4 is 2

3) 10 minus 2 is 8

4) Square root of left extremity which is 7 is…. Well,7 is not a perfect square. So, consider the lower highest perfect square whichis 4. Square root of 4 is 2.

5) Write down 2 as tens digit with both 2 and 8 to get2 possible answers (22 and 28)

6) Now, 2 times its successor which is 3 (2x3) is 6.The left extremity which is 7 is greater than 6. Therefore, the right answermust be the greater option which is 28.

Let’s go for another example: √3969 (square rootof 3969)

1) Disregard the digit next the last digit which is 6 (√39x9)

2) Square root of right extremity which is 9 is 3

3) 10 minus 3 is 7

4) Square root of left extremities which is 39 is….Well, 39 is not a perfect square. So consider the lower highest perfect squarewhich is 36. Square root of 36 is 6.

5) Write down 6 as tens digit with both 3 and 7 to get2 possible answers (63 and 67)

6) Now, 6 times its successor which is 7 (6x7) is 42.The left extremity which is 39 is smaller than 42. Therefore, the right answermust be the smaller option which is 63.

Further example: √5476 (square root of 5476)

1) Disregard the digit next the last digit which is 7 (√54x6)

2) Square root of right extremity which is 6 is…. Wellhere 6 is not a perfect square so it sould be solved from the left extremitieswhich is 54. Now, 54 is not also a perfect square. So consider the lower highestperfect square which is 49. Square root of 49 is 7.

3) Write down 7 as tens digit with the left extremitiy6 and 4 (10-6) to get 2 possible answers (76 and 74)

4) Now, 7 times its successor which is 8 (7x8) is 56.The left extremity which is 54 is smaller than 56. Therefore, the right answermust be the smaller option which is 74.

√625(square root of 625)

1) Disregard the digit next the last digit which is 2 (√6x5)

2) Square root of right extremity which is 5 is…. Wellhere 5 is not a perfect square so, it should be solved from the left extremitieswhich is 6. Now, 6 is not also a perfect square. So consider the lower highestperfect square which is 4. Square root of 4 is 2.

3) Write down 2 as tens digit with the left extremity 5and 5 (10-5) to get 2 possible answers (25 and 25)

4) Now, 2 times its successor which is 3 (2x3) is 6.The left extremity which is 6 is equal to 6. Therefore, the right answer mustbe the greater option which is 25. In this case both options are same. So, theanswer must be 25.

One final example: √13689 (square root of 13689)

1) Skip the digit next the last digit which is 8 (√136x9)

2) Square root of right extremity which is 9 is 3

3) 10 minus 3 is 7

4) Square root of left extremity which is 136 is….Well, 136 is not a perfect square. So, consider the lower highest perfect squarewhich is 121. Square root of 121 is 11.

5) Write down 11 as tens digit with both 3 and 7 to get2 possible answers (113 and 117)

6) Now, 11 times its successor which is 12 (11x12) is 132.The left extremity which is 136 is greater than 132. Therefore, the rightanswer must be the greater option which is 117.

Technique–2 (for larger number):

To be continued …………..

Extracting Cube Roots

2011-12-14T12:27:00.000+06:00

An easy task for the beginner is extracting cube roots from the cubes of 2 digit numbers. For example, given 74088, determine what two digit number, when multiplied by itself once and then multiplied by the number again, yields 74088. One who knows the method will quickly know the answer is 42, as 42³ = 74088.

Before learning theprocedure, it is required that the performer memorizes the cubes of the numbers1-10:

1³ = 1

2³ = 8

3³ = 27

4³ = 64

5³ = 125

6³ = 216

7³ = 343

8³ = 512

9³ = 729

10³ = 1000

To calculate faster also note that,

If perfect cube ends in 0, the cube root of it must end in 0.
If perfect cube ends in 1, the cube root of it must end in 1.
If perfect cube ends in 2, the cube root of it must end in 8.
If perfect cube ends in 3, the cube root of it must end in 7.
If perfect cube ends in 4, the cube root of it must end in 4.
If perfect cube ends in 5, the cube root of it must end in 5.
If perfect cube ends in 6, the cube root of it must end in 6.
If perfect cube ends in 7, the cube root of it must end in 3.
If perfect cube ends in 8, the cube root of it must end in 2.
If perfect cube ends in 9, the cube root of it must end in 9.

Note that every digit corresponds to itself except for 2, 3, 7 and 8 which are just subtracted from ten to obtain the corresponding digit.

The Technique

There are two steps to extracting the cube root from the cube of a two digit number. Say you are asked to extract the cube root of 29791. Begin by determining the one's place (units) of the two digit number. You know it must be one, since the cube ends in 1, as seen above.

The second step is to determine the first digit of the two digit cube root by looking at the magnitude of the given cube. To do this, remove the last three digits of thegiven cube 29~~791~~ (29791 -> 29) and find the greatest cube it is greater than (this is where knowing the cubes of numbers 1-10 is needed). Here, 29 is greater than 1 cubed, greater than 2 cubed, greater than 3 cubed, but not greater than 4 cubed. The greatest cube it is greater than is 3, so the first digit of the two digit cube must be 3.

Therefore, the cube root of 29791 is 31.
Another example:

Find the cube root of 456533.
The cube root ends in 7.
After the last three digits are taken away, 456 remains.
456 is greater than all the cubes up to 7 cubed.
The first digit of the cube root is 7.

The cube root of 456533 is 77.

Further Example: Cube Root of 300763

Divide in to groups of 3 from right = 300 763

Note: The number ends in a 3. Last digit of cube will be 7 if this is a cube without a remainder

Since 7 cubed = 343 and 6 cubed = 216, the left most group of 300 is between them and we must use the smaller, or 6.

The answer is 67

This method works up to 1,000,000 for true cubes

Squaring any 2-digit numbers

2011-12-10T22:49:00.003+06:00

Squaring any 2-digit numbers mentally is actually the fastest way and very interesting to calculate. It's a 4 step simple technique.

Step 1: Square each digit and write a partial answer.
Step 2: Multiply the two digits of the number you’re squaring together.
Step 3: Double this product; then add a 0 to it.
Step 4: Add both the findings from step 1 and step 3.

Let’s say you want to square 58.
First, square each digit and write a partial answer. 5×5 = 25. 8×8 = 64.
Write down 2564 to start.
Then, multiply the two digits of the number you’re squaring together, 5×8=40.

Now, double this product: 40×2=80, then add a 0 to it, getting 800.

Finally, add 800 to 2564 to get 3364.

This is pretty complicated so let’s do more examples.

32² = 32 × 32

The first part of the answer comes from squaring 3 and 2.

3×3=9

2×2 = 4

Write down 0904. Notice the extra zeros. It’s important that every square in the partial product have two digits.

Multiply the digits, 2 and 3, together and double the whole thing. 2×3x2 = 12.

Add a zero to get 120. Add 120 to the partial product, 0904, and we get 1024.

56×56. The partial product comes from 5×5 and 6×6. Write down 2536.

5×6x2 = 60. Add a zero to get 600.

56×56 = 2536+600 = 3136.

One more example: 67×67. Write down 3649 as the partial product.

6×7x2 = 42×2 = 84. Add a zero to get 840.

67×67=3649+840 = 4489.