From Newsgroup: sci.logic

Conquer the Binary Tree

.
/ \
0 1
/\ /\
0 1 0 1
/\ /\ /\ /\
...

The complete infinite Binary Tree consists of nodes representing bits
(binary digits 0 and 1) which are indexed by non-negative integers and connected by edges such that every node has two and only two child
nodes. Node number 2n + 1 is called the left child of node number n,
node number 2n + 2 is called the right child of node number n. The set
{a_k | k ree rao_0} of nodes a_k is countable as shown by the indices of the nodes.

To play the game Conquer the Binary Tree you start with one cent. For
one cent you can buy an infinite path of your choice in the Binary Tree.
For every node covered by this path you will get a cent. For every cent
you can buy another path of your choice. For every node covered by this
path (and not yet covered by previously chosen paths) you will get a
cent. For every cent you can buy another path. And so on. Since there
are only countably many nodes yielding as many cents but uncountably
many paths requiring as many cents, the player will get bankrupt before
all paths are conquered. If no player gets bankrupt, the number of paths cannot surpass the number of nodes.

Note: If set theory is right, then most paths that you can buy do not
contain new nodes.

Regards, WM
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