A square with 1,536 soldiers are not possible, i.e. the square root of 1,536 is about 39.191, which means that 1,536 is not a square a number. In order for it to be a square number, we have to round up 39.191 (because it says "how many more soldiers") to 40 and 40×40=1,600 and 1,600-1,536, so we need 64 more soldiers.
The greatest 4 digit number is simple. If the square root of 10,000 is 100, than, one can assume that because 100-1=99, the resulting answer of 99×99 is less than 10,000, meaning that is is 4 digits. 99×99=9,801.
You need Pythagorean theorem for this.
a² + b² = c², where c is the hypotheneus and a and b are the sides of the right triangle.
24² + x² = 26²
x² = 676-576
x = ±√100
x = 10 (-10 is rejected because length of triangle cannot be a negative number)
