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?
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.
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
The table below shows some square numbers and the corresponding
numbers on the seven-clock (representing these numbers modulo 7).
This works like the days of the week.
For example we say 25 = 4 (mod 7) because when counting up to 25
around the clock you get to the number 4. To avoid lots of counting
simply divide 25 by 7 to get 3 remainder 4. Modulus (or clock)
arithmetic uses the remainders when one number is divided by
Take the number 11 and calculate 1 2, 2 2,
up to 10 2 modulo 11.
Take the number 13 and calculate 1 2, 2 2, up
to 12 2 modulo 13.
What do you notice? What else can you say?