I put eggs into a basket in groups of 7 and noticed that I could
easily have divided them into piles of 2, 3, 4, 5 or 6 and always
have one left over. How many eggs were in the basket?
How many integers between 1 and 1200 are NOT multiples of any of
the numbers 2, 3 or 5?
A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the cogwheel A as the wheels rotate.
Tabs from St. Mary's School noticed
Esther Shindler made similar observations:
Pippa and Sophie from The Mount School in York
produced a table of results and added some comments
For prime numbers, such as the number 5, all possible step sizes
will work. All the number of step sizes are even.
Yanqing from Devonport High School for Girls
produced a useful summary of her results:
When the circle has 8 dots, I found that you can hit all the
points with steps of 1, 3, 5 and 7 points. Step sizes of 2, 4 and 6
misses some points.
When the circle has 9 dots, step sizes of 1, 2, 4, 5, 7 and 8
will hit all the points, and 3 and 6 misses some points.
In a circle of 10 dots, you can hit all the points with step
sizes of 1, 3, 7 and 9. 2, 4, 5, 6 and 8 misses some points.
I found a pattern in these results. All the step sizes that
misses some points share a factor other than 1 with the circle
size. From these, I have concluded that as if the highest common
factor between the number of dots on the circle and the step size
is 1, you can hit all the points.
With a circle of 5 dots, you can hit all the points with any
number of steps. Other circles that do this are 3, 7, 11, 13, 17
etc. These are all prime numbers. This fits the pattern: prime
numbers do not have any factors larger than 1, and so all the step
sizes can hit all the points.
The number of steps that will ensure that all points are hit are
the same as the number of points in the circle. This is because you
will need to go to all the points, and this would mean you would
need a step for each point.
Stephen from Pike County High School came to a
Well done to you all.