Pebbles

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

Happy Numbers

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

Intersecting Circles

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

Logoland - Sequences

Stage: 3 Challenge Level:

Below are two diagrams concerned with a sequence of squares

Detailed pattern of squares

Part 1

Full pattern of squares

Part 2

Both are based on the following procedure:

TO SQUARE :S :N
IF :S < 2 [ STOP]
REPEAT 2 [ REPEAT :N [ FD :S RT 90 FD :S LT 90] LT 180]
SQUARE :S/3 :N * 3
END

try SQUARE 120 3

Why not experiment with the procedure? Vary the value of S or N. Why not create a sequence of embedding hexagons? Pentagons? Pentagrams?

Alternatively:

Why not create your own fractals using recursion?

Explore further these well known procedures:

TO FIB :X :Y
PR :Y WAIT 60
FIB :Y :X + :Y
END

try FIB 1 1 or FIB 2 4 or FIB 3 5..............

or this:

TO SQ :X :A
PR :A
WAIT 60
SQ :X + 1 :A + 2 * :X - 1
END

try SQ 1 -2 or SQ -2 -2 or SQ 1 0....................

Devise your own procedures for generating sequences