Squares and Some More Squares

okay let me see if i can exlain this....its kinda hard

okay so we know that the sum of the digits is 1 more than a perfect square. so sum can only be 5, 10,17,26,37,50,65,82....from this 5 is too small and 37, 50, 65 and 82 are too large i.e. the number will be huge and problem will be almost impossible.

now keeping that in mind , lets consider its a 4 digit number

therefore the last 2 digits and the last digit should also be perfect squares (given in question). hence the last 2 digits can only be 81, 64, 25,36,49

also as mentioned in the question last 3 digits should also be perfect squares. but no 3 digit number that ends with 81 or 64 is a perfect square hence they can be rejected.

lets check 25 only 225 and 625 are three digit numbers that end with 25. therefore we can say that 225 and 625 may be the last three digits of the given number.let the last digit be 'x'

now as mentioned above their sum can only be 10,17,26 so

x+2+2+5 = 10 => x=1

therefore the number we get is 1225 which satisfies all the conditions

some people say that 0 is a perfect square some not. if you take that into account and also that 1 is also a perfect square then the correct answer will be 100. Now if 0 is not considered a perfect square then the correct answer is 1225 as the others have commented!!

all numbers less than 1000 does not satisfy, sqrt(1000) is arround 31 so trying squares of 32,33,34 does not satisfy and square of 35 satisfies = 1225

Let's assume number of digits is 3. Then last number could be 1,4,9. Now the last two digits could be 81,64,49. When we remove last digit we get digits ending with 8,6,9(minus 1 should give a perfect sq). So we get 264. But sum of digits is 12 so it doesn't follow the condition.

Now let's assume number of digits is 4. Then last two digit could be 16,25,36,49,64,81. The last three digits is one of 225 or 625.(merge two conditions) As whole number should be a perfect sq. 1225 satisfies the condition.

The Number, number except first digit, number except first two digits are all squares.Which implies that the smallest of the digits (the ones at the units, tenths and higher places) are repeated in the sequence of perfect squares. Now, noticing the sequence of perfect squares 1,4,9,16,25,... its easy to notice that the number 25 (5-> unit, 2->tenth) is repeated most frequently. Checking, this sequence gives 1225, the forth number in the sequence and also the correct answer.

It should be 1225. 225 is perfect square, 25 is perfect sq, 122 is 1 more than a perfect sq.