Learn how to mentally calculate square roots and amaze your friends!

Performance

By learning this simple system you will be able to instantly calculate the square root of the spectator's number.

Note that calculating square roots is rather more difficult than either cube roots or fifth roots - you should learn these two procedures first.

Ask the spectator to choose any whole number less than 100 and, using a calculator, to square it by multiplying the number by itself.

The spectator then calls out the answer, and you instantly reveal the original number (i.e., the square root).

Note that this method will not work with squares of numbers greater than 99. Also the method only works when the square root is a natural number (integer).

Step 1: Learn the Squares of 0 to 9

To master the system you must learn by heart the squares of numbers 0 to 9, which are shown in the table below. You also need to consider the last digit of each square.

Number / Square / Last Digit
0 / 0 / 0
1 / 1 / 1
2 / 4 / 4
3 / 9 / 9
4 / 16 / 6
5 / 25 / 5
6 / 36 / 6
7 / 49 / 9
8 / 64 / 4
9 / 81 / 1

Note how the last digits for the squares of 1 and 9 are both 1.

Note how the last digits for the squares of 2 and 8 are both 4.

Note how the last digits for the squares of 3 and 7 are both 9.

Note how the last digits for the squares of 4 and 6 are both 6.

Step 2: Determine the Square Root

Ignore the last two digits of the number called out by the spectator and choose the memorised square which is just lower (or equal) to the remaining number. The corresponding square root is the first digit. of your answer

Now consider the last digit of the number called out by the spectator. If this is 0 or 5, then you immediately know that the last digit of your answer is also 0 or 5.

In all other cases, the last digit of the number called out will indicate two possible values for the last digit of the square root. For example, if the last digit is 9, then the square root may end in either 3 or 7.

To determine whether the lower or higher value should be taken, multiply the first digit of your answer by one greater than itself. If this is greater than the first part of the number called by the spectator (i.e., ignoring the last two digits), then the last digit of your answer is the lower of the two possible values. Otherwise the last digit is the higher value.

For example if the number called is 2809, the square root could be either 53 or 57. Since 5 x 6 = 30, and 28 is less than 30, the answer is 53.

Examples

Spectator Calls / Ignore Last 2 / Lower Square / First Digit (a) / Last Digit / a(a+1) / Square Root
144 / 1 / 1 / 1 / 2 or 8 / 2 / 12
784 / 7 / 4 / 2 / 2 or 8 / 6 / 28
1369 / 13 / 9 / 3 / 3 or 7 / 12 / 37
2401 / 24 / 16 / 4 / 1 or 9 / 20 / 49
2809 / 28 / 25 / 5 / 3 or 7 / 30 / 53
3600 / 36 / 36 / 6 / 0 / 60
5041 / 50 / 49 / 7 / 1 or 9 / 56 / 71
7396 / 73 / 64 / 8 / 4 or 6 / 72 / 86
8836 / 88 / 81 / 9 / 4 or 6 / 90 / 94

Step 3: Practice, Practice, Practice!

Before you try this out on your friends, you should practice until you can calculate the square root instantly and without error. If you can't do this, then practice some more! To practice and assess your ability, you can use the test below.

Practice on