Problem: Calculate 572
Method: The Square 50 method allows you to quickly find the square of any number that is around 50 (30 to 70). First we determine just how close the number is to 50. To do this we subtract 50 from the number that we want to square (note that this difference will be positive if our number is greater than 50 and negative if our number is less than 50). Then we add this value to 25. The result will represent the beginning digits of our answer.
The trailing two digits can be determined by squaring the distance that we determined above (i.e., compute the square of the distance from 50). If the square of the distance is a single digit number then we include a zero digit at the start so that we have two digits. If the square of the difference is a three digit number then the first digit is considered to be overflow and we add this overflow to the leftmost digits that we already determined.
In our problem, we observe that 57 is 7 more than 50. Next we add this value of 7 to 25 and we get 32. So, 572 begins with 32. Now to find the last two digits of the product we square 7 and get 49. So we get 572 = 3249.
Problem: Calculate 532
Method: In this problem we see that the distance from 50 is 3. We add 3 to 25 to determine that the result will begin with the digits 28. Next we square the distance and get a value of 9. Since this is a single digit number we prefix a zero to get 09. Now, we can see that 532 = 2809.
Problem: Calculate 622
Method: In this problem we see that the distance from 50 is 12. We add 12 to 25 to determine that the result will begin with the digits 37. Next we square the distance and get a value of 144. Since this is a three digit number we have an overflow condition with a value of 1 and we need to adjust our starting digits by adding 1 to 37 to get 38. Now, we can see that 622 = 3844.
Problem: Calculate 462
Method: In this problem we see that the distance from 50 is -4. We add -4 to 25 to determine that the result will begin with the digits 21. Next we square the distance and get a value of 16, so our result will end with 16. Now, we can see that 462 = 2116.