Properties of Squares

Integer (n)
Square Root (√n)
Square (n2)
Square (n+1)2
Ratio   (n+1)2 / (n2)
Square  (n2) as sum of odd no. from 1 to  
Square as sum of consecutive  triangular no.Tn+Tn-1 +
Centered square n2+(n-1)2
Centered Octagonal Square (2n-1)2
sum of square of integers up to n Σ n2 (n=1 to n)
n-sum of maximum 4  squares : + ++ + *
Nearest n which is a square ,
If a number ends with 2,3,7,8, its square root is not an integer.
If a number ends with 25, its square root is an integer ending with 5.
If a number is a square and has a zero, the no. of last zeroes have to be 2 or multiple thereof.
If a no. is a square and ends with 6,its second digits are odd (1,3,5,7,9) and the square root ends with 4 or 6.
If a no. is a square and ends with other than 6, second digits are invariably even i.e. (0,2,4,6,8)
If a no. is a square & ends with 1, its root ends with (1,9)
If a no. is a square & ends with 9, its root ends with (3,7)
If a no. is a square & ends with 4, its root ends with (2,8)
If a no. is a square & ends with 6, its root ends with (4,6)
Ratio of 2 consecutive squares of (n+1) and n is (1+1/n)2. Maximum value of ratio is 4 and minimum is 1
If square number is even, the square root is also even and if square is odd, square root is also odd.
A square number is the sum of 2 consecutive triangular numbers. Triangular numbers  are no. of objects that can form an equilateral triangle. Formula of a triangular no. is n(n+1) / 2. No. are 1,3,6,10,15,21,28,36, .....The series is additive analog of factorial. Sum of n triangular no. starting with 1 is given by n(n+1)(n+2)/6.
Sum of 2 consecutive squares is a centered square given by the formula C4,n= n2+(n-1)2The series is 1,5,13,25,41,61,.....
Every odd square is a centered octagonal number given by  (2n-1)2=8 Tn-1   + 1 where Tn-1 is the triangular no. corresponding to n-1. The series is 1,9,25,49,81,.....Last digits of Octagonal numbers follow 1-9-5-9-1 pattern.
The sum of the squares of n natural numbers starting from 1 is  n(n+1)(2n+1)/6

The King & the Mathematician : It was the 1st day of the month of April. The king offered his court mathematician 1 gold coin and asked him a question. he would be given coins on every odd date of the month and the no. of coins will be same as the date.How many coins he shall have at the end of month. Prompt answer will ensure coin supply for the whole month. Failure to answer shall result in return of the gold coin given. The mathematician,no sooner the question was finished, answered 225. The king was pleased. How didculate so quickly? The sum of first n odd numbers is equal to the square of the no. of terms. Here no. dates were ((29-1)/2)+1 =15 and sum of odd no. from 1 to 29 is 15*15=225. or Σ (2k-1) = n2 (k=1 to n)

Joseph Louis Lagrange in 1770 proved that any positive  integer can be expressed as a sum of not more than 4 positive squares.* is the residue which indicates a number may require to be expressed in terms of squares more than 4.Study-->331,99971
The square of a number which contains all digits from 1 to 9 and is a palindrome is (111111111)2 =12345678987654321
Undulating squares are those whose digits follow a pattern (ababab). Exa.-- 2642 =69696
Largest perfect square that has all digits from 0 to 9. -------------------------990662 =9814072356

Smallest perfect square that has all digits from 0 to9. -------------------------320432 =1026753849

Joseph Madachy discovered numbers that are equal to their halves squared when they are split apart  like 1233 = 122 + 332

8833 = 882 + 332    ---- 5882353 = 5882 + 23532

There are numbers which are square no. and on reversing the digits, the consequent no. is also a square and on multiplication of the no. and its reversed no. the generated no. is also a square. Exa-- 169*961=4032 =162409    & 1089*9801=32672=10673289
 x! +1=y2has x=4,5,7 This is known as Brocard's problem.

No. of digits in Numbers & their Squares

 Range   (n1 - n2) no. of digits in n    (x) no. of digits in n2 (y)  y / x  n2 /n1
1 - 3  1 1 1 3
4 - 9  1 2 2 2.25
10-31  2 3 1.5 3.1
32-99  2 4 2.0 3.09375
100-316 3 5 1.66 3.16
317-999 3 6 2.0 3.151419
1000-3162 4 7 1.75 3.162
3163-9999 4 8 2.0 3.161239
10000-31622 5 9 1.8 3.1622
31623-99999 5 10 2.0 3.162223
100000-316226 6 11 1.833 3.16226
316227-999999 6 12 2.0 3.162282