Prize Puzzle Competition 29

Closing date is 30th June 2001

85. A 4 by 4 grid, like that shown, is to be cut into 4 pieces. Cutting may only be done along the lines and the 4 pieces must be identical to each other in both size and shape.
To see if pieces match each other they may be turned around but not turned over.
In how many different ways can this be done?
4 b 4 grid

86. My office is on the 3rd floor and, though there is a lift, I like using the big circular staircase. I go up the stairs 2 at a time (to get more exercise) but am annoyed that there is always just 1 stair left over when I get to my landing.
One day, in a burst of energy I went up the stairs 3 at a time, and still had 1 stair left over!
By counting (certainly not by doing it!) I established that even if I could go up 4, 5 or 6 stairs at a time there would still be 1 left over.
Worse yet, no matter how many stairs I did at a time (except for 1 or the lot) there would always be some left over.
How many stairs are there to my office?

87. A new recreation space is being laid out. It will be oblong in shape and 55 metres long, but no width has yet been fixed.
A square swimming-pool is to be placed in the middle of the length, and is to fill the full width of the space, leaving a paved area at each end.
To get the largest possible paved area,
what should be the width of the recreation space?
Recreation space

The winner of Competition #28 was Steve Morris of Sandbach, Cheshire.
The answers were:

82. The digit 1 appears 301 times.
83. The start must be at B or F and the finish at F or B.
There are many possibilities, but each letter must appear twice.
84. n = 664 gives 2086 (+0.0175 . . .)