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Formulas and tips covering multiple ranges.
First note the squares of 25, in order to get a ballpark figure of the magnitude of the numbers.
This will help to trim the numbers later on, if they seem too large.
202=400, 212=441, 222=484, 232=529, 242=576, 252=625
Case 1:
Step 1: Add the units digit to the whole number.
Step 2: Square the units digit, and include it to the right of the result in step 1.
Example: 122
12+2=14
22=4
144
Case 2:
Step 1: Add the units digit to the whole number.
Step 2: Square the units digit, and include it to the right of the result in step 1.
Example: 152
15+5=20
52=25
2025
In this case 2,025, is too large; it is much greater than the square of 25 (625).
Remove the zero for the correct answer: 225.
Case 3:
Step 1: Add the units digit to the whole number.
Step 2: Square the units digit, and include it to the right of the result in step 1.
Example: 142
14+4=18
42=16
1816
In this case 1,816, is too large; it is much greater than the square of 25(625)
Carry over the one and add it to the eight for the correct answer: 196.
Step 1: Take any two numbers equidistant from the number (they don't have to lie on the range 11-22). It's convenient to locate one number that has a zero units digit.
Step 2: Multiply the two numbers.
Step 3: Add the square of the distance of one of the equidistant numbers.
Example: 172
14×20=280
32=9
280+9=289
Step 1: Subtract 25 from the number; this number is 'x'.
Step 2: Subtract the previous result from 25; this number is 'y'.
Step 3: Substitute x and y in the following mathematical expression:
100x + y2
Example: 282
28−25=3 (x)
25−3=22 (y)
(100)3+222=784
Step 1: Find the difference between the number and 50. Over 50 is positive difference, and below 50 is negative.
Step 2: Add 25. This will take the leftmost position in the answer.
Step 3: Square the difference calculated earlier. This will take the rightmost position in the answer.
Example: 432
Difference between 50 and 43 is −7.
25-7=18 (leftmost digits)
(-7)2=49 (rightmost digits)
1,849
This formula calculates a square when the previous square is known.
y + x + x2 = y2
Example: 1012
y=101
x=100
x2=10,000 (known).
101+100+10,000=10,201
Elevator method.
Step 1: Note the difference between the number and the closest zero units digit number. If the number is eighteen, for example, twenty is closest, and the difference is two.
Step 2: If the zero units digit number was greater, go down by the same amount. Vice versa, if the zero digit number was lesser, go up.
Step 3: Step 1 × step 2.
Step 4: Square the step 1 difference.
Step 5: Step 3 + step 4.
Example 1: 182
Go up to 20 (difference = 2)
Go down 2, from 18 to 16
20×16=320
22=4
320+4=324
Example 2: 482
Go up to 50 (difference = 2)
Go down 2, from 48 to 46
46×50=2,300
22=4
2,300+4=2,304
Example 3: 632
Go down to 60 (difference = 3)
Go up 3, from 63 to 66
60×66=3,960
32=9
3,960+9=3,969
Step 1: Add 25 to the units digit.
Step 2: Affix the square of the former units digit to the right of the result in step 1.
Step 3: Estimate a ballpark figure; zeros may need to be added or trimmed from the final result.
Example: 522
2+25=27
22=4
274
Using 50 as a benchmark: 502 = 2,500.
The 274 answer is therefore too small a number.
Add a zero to the left of 4.
2704
Step 1: Take the leftmost digit(s) and multiply with the next highest integer.
Step 2: Attach 25 on the right; 25 will occupy the tens and units digits.
Example 1: 452
4×5=20
2025
Example 2: 1452
14×15=210
21,025