Solutions to Problems 1361 - 1370

Q1361 Find a six-digit number which can be split into three two-digit squares and also into two three-digit squares.  (The first digit of a number cannot be zero.)
SOLUTION Find a six-digit number which can be split into three two-digit squares and also into two three-digit squares.  (The first digit of a number cannot be zero.)

The number must begin with a three--digit square whose first two digits also form a square.  So we seek a three--digit square of the form$$\def\\#1{{\rm#1}} 16\\X\quad\hbox{or}\quad25\\X\quad\hbox{or}\quad36\\X\quad\hbox{or}\quad49\\X\quad\hbox{or}\quad64\\X\quad\hbox{or}\quad81\\X\quad;$$ the possibilities are $169$, $256$ and $361$.  The last of these digits must begin a two--digit square, which rules out $169$.  The remaining options for our six--digit number are $$2564{\rm X}{\rm Y}\quad\hbox{and}\quad3616{\rm X}{\rm Y}\ .$$ Now $4{\rm X}{\rm Y}$ is a three--digit square beginning with $4$, and so we have ${\rm X}{\rm Y}=00,41$ or $84$; the first is ruled out by the conditions of the problem and the others are not squares.  The only answer to the problem is $361625$.

Q1362 Sandy leans a ladder against a wall in order to clean the gutter running along the top of the wall.  Sandy is worried that the foot of the ladder is going to slip away from the wall and therefore ties a tightly stretched string between the middle of the ladder and a nail which is located directly below the top of the ladder, at the point where the floor meets the wall.  Assuming that the floor is perfectly horizontal and the wall is perfectly vertical, how much is this going to help?

SOLUTION If the foot of the ladder slips away from the wall then the middle of the ladder is always the same distance from the nail.  (Why?  Because the angle between the wall and the floor is a right angle, so the line from the nail to the middle is always a radius of the circle having the ladder as diameter.)  So connecting these two points by a string is not going to help at all!!