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Whole Number Dynamics I

The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.

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Whole Number Dynamics II

This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.

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Whole Number Dynamics III

In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.

Odd Stones

Age 14 to 16 Challenge Level:

$27$ stones are distributed between $3$ circles

On a "move" a stone is removed from two of the circles and placed in the third circle.

So, in the illustration, if a stone is removed from the $4$ and the $10$ circles and added to the $13$ circle, the new distribution would be $3$ - $9$ - $15$

Check you can turn $2$ - $8$ - $17$ into $3$ - $9$ - $15$ in two "moves"

Here are five of the ways that $27$ stones could be distributed between the three circles :

$6$ - $9$ - $12$

$3$ - $9$ - $15$

$4$ - $10$ - $13$

$4$ - $9$ - $14$

$2$ - $8$ - $17$

There is always some sequence of "moves" that will turn each distribution into any of the others - apart from one.

Identify the distribution that does not belong with the other four.

Can you be certain that this is actually impossible rather than just hard and so far unsuccessful?