Sol: Remember that: If square of a number ends in same unit digit as that of number, then that unit digit can be 0, 1, 5 or 6 only. So possible values for G are 0, 1, 5 or 6. Now by checking you can easily negate that G can't be 0 as GOG will not remain a three digit number. It can't be 1 either as in this case the three digit number will be 1O1 and it'll be perfect square only when O is 2 i.e. T is 1 which is not possible as T and G are "DISTINCT" single digit positive integers. Similarly you can check for 5 also that no number of the form 5O5 is a perfect square.
Only possible case is 26² = 676 i.e. O = 7.

@vikas_saini can u plz explain 2nd and 3rd+4th step again. 2nd step:what is -1-what are we reducing in it
3rd step: 3rd+4th step doesnt giv value in 2nd step. what are we missing in it. 3rd+4th= 2T-2

Let us number the vertices from 1 – 6, there are three different types of triangles that can be formed.Type 1 : (1, 2, 3) [ this is same as (2, 3, 4), (3, 4, 5) … etc)Type 2 : (1, 2, 4) [ this is same as (1, 2, 5) (2, 3, 5) (3, 4, 6)]Type 3 : (1, 3, 5) and (2, 4, 6) have same shape and area
So there will be three triangles of different areas that can be formed from a regular hexagon.There are 6 triangles of Type 1, 12 of type 2 and 2 of type 3.Adding these we get 6 + 12 + = 20 triangles.
Total number of triangles possible = 6C3 = 20

@Naman-Jain-0
___ | ___ | ___ | ___ | ___ | ___ | ___ | ___There are a total of 8 steps represented by (___) and let the dash ('|') represent if he is stopping at the previous step.
There are 7 such dashes. Each dash can take a 0 or 1. 0 indicates, he is stopping at the step immediately before it.
As each dash can take a 0 or 1, the number of ways is 128.And as the maximum steps u can take is 6, the cases where he takes 7 steps at a time - given by (1,0,0,0,0,0,0) and (0,0,0,0,0,0,1)- and 8 steps at a time - given by (0,0,0,0,0,0,0)- are eliminated.
Therefore it is 125.

Can start by understanding that the numbers will be close to the cube root of 190740 which should lie between 50 and 60 (between 125000 and 216000). Can start with the smallest prime numbers and check. It will be satisfied for 53 * 59 * 61. Total of 53 + 59 + 61 = 173.