Squaring 3 digit numbers!

Squaring 3 digit numbers is kind of the penultimate feat in a mathemagician’s bag of tricks. Just short of being able to square 4 digits, 5 digits and multiplying two random 5 digit numbers.

To square a 3 digit number, here are the steps:

Step 1: Given a 3-digit number, round it up or down to the nearest hundred. For example, 314 would go down to 300. 567 would go up to 600. Additionally, note the amount you take to move the number to the nearest hundred, and move the given number by that same amount in the opposite direction. For example, when you moved 314 down to 300, you would also generate another number 328 by moving 314 14 units up. Similarly you moved 567 33 units up to 600. Now move it 33 units down to 534.

Step 2: Multiply that two new numbers that you generated in step 1. For example, you would multiply 300 by 328 and 600 by 534.

Step 3: Remember that amount by which you had to move the given number to make it the nearest hundred? Square it.

Step 4: Add the square you got to the product in step 2.

Example: $717^2$
Step 1: move 17 down to reach 700 and simultaneously go 17 up to reach 734.

Step 2: Multiply 700 x 734 to get 513800

Step 3: Do $17^2$ to get 289

Step 4: Add 289 to 513800 to get 514089.

Now there are certain complications which you might have noticed if you were working along the problem as you read it. Assuming you are an intermediate at this and were able to perform step 2, it seems to many that it is slightly difficult to keep 513800 in mind whilst simultaneously working out $17^2$ . Experience has taught me that that by the time you complete step 3, you forget what you got in step 2. There are two remedies for this:

Option 1: Memorize the squares up to 50 (You won’t need them past 50. why?)
OR
Option 2: Use shortcuts for computing the squares.

Now, there are several famous shortcuts out there for calculating squares of two digits. In another entry, I shall be talking about a different shortcut I thought of that I could use and find it much faster and easier than the traditional ones. By employing shortcuts for step 3, it becomes easier to hold the product of step 2 in your mind and thus allows you to perform this calculation in your head.