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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.

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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?

Clock Squares


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.

Square numbers in base ten 1 4 9 16 25 36 49 64 81 100
Square numbers modulo 7 1 4 2 2 4 1 0 1 4 2

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 another.

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?